Class 10th Mathematics Gujarat Board Solution
Exercise 9.1- In ΔABC, m∠A = 90. If AB = 5, AC = 12 and BC = 13, find sinC, cosC, tanB, cosB,…
- In ΔABC, m∠B = 90. If BC = 3 and AC = 5, find all the six trigonometric ratios…
- If cosA = 4/5 find sinA and tanA.
- If cosec θ = 13/5 find tan θ and cos θ.
- If cosB = 1/3 , find the other five trigonometric ratios.
- In ΔABC, m∠A = 90 and if AB : BC = 1 : 2 find sinB, cosC, tanC.
- If tan θ = 4/3 , find the value of 5sintegrate heta +2costheta /3sintegrate heta…
- If sec θ = 13/5 find the value of 5sintegrate heta +3costheta /5costheta…
- If sinB = 1/2 prove that 3cosB - 4cos^3 B = 0.
- If tanA = √3, verify that (1) sin^2 A + cos^2 A = 1 (2) sec^2 A - tan^2 A = 1…
- If cos θ = 2 root 2/3 , verify that tan^2 θ - sin^2 θ = tan^2 θ⋅ sin^2 θ…
- In ΔABC, m∠B = 90, AC + BC = 25 and AB = 5, determine the value of sinA, cosA…
- In ΔABC, m∠C = 90 and m∠A = m∠B, (1) Is cosA = cosB? (2) Is tanA = tanB? (3)…
- If 3cotA = 4, examine whether 1-tan^2a/1+tan^2a cos^2 A - sin^2 A.…
- If pcot θ = q, examine whether psi ntheta -qcostheta /psi ntheta +qcostheta =…
- State whether the following are true or false. Justify your answer: (1) sin θ =…
Exercise 9.2- cos60 = 1 - 2sin^2 30 = 2cos^2 30 - 1 = cos^2 30 - sin^2 30 Verify:…
- sin60 = 2sin30 cos30 Verify:
- sin60 = 2tan30/1+tan^230 Verify:
- cos60 = 1-tan^230/1+tan^230 Verify:
- cos90 = 4cos^3 30 - 3cos30 Verify:
- sin30+tan45-cosec60/sec30+cos60+cot45 Evaluate:
- 5cos^260+4sec^230-tan^245/sin^230+cos^230 Evaluate:
- 2sin^2 30 cot30 - 3cos^2 60 sec^2 30 Evaluate:
- 3cos^2 30 + sec^2 30 + 2cos0 + 3sin90 - tan^2 60 Evaluate:
- In ΔABC, m∠B = 90, find the measure of the parts of the triangle other than the…
- In a rectangle ABCD, AB = 20, m∠BAC = 60, calculate the length of side bar bc…
- If θ is measure of an acute angle and cosθ = sinθ, find the value of 2tan^2 θ +…
- If α is measure of acute angle and 3sinα = 2cosα, prove that (1-tan^2alpha…
- sin(A + B) = sinA cosB + cosA sinB, If A = 30 and B = 60, verify that…
- cos(A + B) = cosA cosB - sinA sinB If A = 30 and B = 60, verify that…
- If sin(A - B) = sinA cosB - cosA sinB and cos(A - B) = cosA cosB + sinA sinB,…
- State whether the following are true or false. Justify your answer: (1) The…
Exercise 9.3- cos18/sin72 Evaluate:
- tan48 — cot42 Evaluate:
- cosec32 — sec58 Evaluate:
- cos70/sin20 + cos59 ⋅ cosec31 Evaluate:
- sec70 sin20 — cos20 cosec70 Evaluate:
- cos(40— theta) — sin(50 + theta) + cos^240+cos^250/sin^240+sin^250 Evaluate:…
- cos70/sin20 + cos55cosec35/tan5tan25tan45tan65tan85 Evaluate:
- cot12 ⋅ cot38 ⋅ cot52 ⋅ cot60 ⋅ cot78 Evaluate:
- sin18/cos72 + √3 (tan10 tan30 tcm40 tan50 tan80- Evaluate:
- cos70/sin20 + sin22/cos68 - cos38cosec52/tan18tan35tan60tan72tan55 Evaluate:…
- sin48 sec42 + cos48 cosec42 = 2 Prove the following:
- sin70/cos20 + cos6c^20/sec70 — 2cos70 cosec20 = 0 Prove the following:…
- Prove the following:
- cos (90-a) sin (90-a)/tan (90 - phi) = sin^2a Prove the following:…
- Express the following in terms of trigonometric ratios of angles having measure…
- For ΔABC, prove that (1) tan (a+c/2) = cot b/2 , (2) cos (b+c/2) = sin (a/2)…
- If A + B = 90, prove that root tanatanb+tanacotb/sinasecb = seca
- If 3 θ is the measure of an acute angle and sin30 = cos(θ — 26), then find the…
- If 0 θ 90, θ, sinθ = cos30, then obtain the value of 2tan^2 θ — 1.…
- If tanA = cotB, prove that A + B = 90, where A and B are measures of acute…
- If sec2A = cosec(A — 42), where 2A is the measure of an acute angle, find the…
- If 0 θ 90 and secθ = cosec60, find the value of 2cos^2 θ — 1.
Exercise 9- cos^2theta + 1/1+cot^2theta = 1 Prove the following by using trigonometric…
- 2sin^2 θ + 4sec^2 θ + 5cot^2 θ + 2cos^2 θ — 4tan^2 θ — 5cosec^2 θ = 1 Prove the…
- 1/1+costheta + 1/1-costheta = 2costheta c^2theta Prove the following by using…
- Prove the following by using trigonometric identities:
- root 1-sintegrate heta /1+sintegrate heta = sectheta -tantheta Prove the…
- sectheta +tantheta /costheta c theta +cottheta = costheta c theta -cottheta…
- cottheta +cosectheta -1/cottheta -cosectheta +1 = cosectheta +cottheta Prove the…
- (sinθ + cosecθ)^2 + (cosθ + secθ)^2 = 7 + tan^2 θ + cot^2 θ. Prove the following…
- 2sec^2 θ — sec^4 θ — 2cosec^2 θ + cosec^4 θ = cot^4 θ — tan^4 θ. Prove the…
- (sinθ — secθ)^2 + (cosθ — cosecθ)^2 = (1 — secθ ⋅ cosecθ)^2 . Prove the…
- Prove the following by using trigonometric identities:
- tantheta -cottheta /sintegrate heta costheta = sec^2theta -costheta c^2theta =…
- sectheta -tantheta /sectheta +tantheta = 1-2sectheta tantheta +tan^2theta Prove…
- root sec^2theta +costheta c^3theta = tantheta +cottheta Prove the following by…
- Prove the following by using trigonometric identities:
- tantheta /1-cottheta + cottheta /1-tantheta = 1+tantheta +cottheta = 1+sectheta…
- sin^4 θ - cos^4 θ = sin^2 θ - cos^2 θ = 2sin^2 θ - 1 = 1 - 2 cos^2 θ. Prove the…
- tan^2 A — tan^2 B = Prove the following by using trigonometric identities:…
- 2(sin^6 θ + cos^6 θ) — 3(sin^4 θ + cos^4 θ) + 1 = 0 Prove the following by…
- If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p^2 — 1) = 2p.…
- If tanθ + sin = a and tanθ — sinθ = b, then prove that a^2 — b^2 = 4 root ab…
- acosθ + bsinθ = p and asinθ— bcosθ = q, then prove that a^2 + b^2 = p^2 + q^2 .…
- secθ + tanθ = p, then obtain the values of secθ, tanθ and sinθ in terms of p.…
- sec38/cosec52 + 2/root 3 ⋅ tan17 tan38 tan60 tan52 tan73 — 3(sin^2 32 + sin^2…
- - cottan (90 - theta) + cos6ctheta sectheta (90 - theta)
- If sinA + cosA = √2 sin(90—A), then obtain the value of cotA.
- If cosecθ = √2, then find the value of 2sin^2theta +3cot^2theta /4tan^2theta…
- If tantheta = 8/15 then evaluate (1+sintegrate heta) (2-2sintegrate…
- If costheta = b/root a^2 + b^2 , 0 θ 90, find the value of sinθ and tanθ.…
- If θ is the measure of an acute angle such that bsinθ = acosθ, then…
- Which of the following is correct for some 0 such that 0 ≤ θ 90?A. 1/sectheta…
- If tantheta = 1/root 5 , then cosec^2theta -sec^2theta /cosec^2theta…
- If tan^2theta = 8/7 , then the value of (1+sintegrate heta) (1-sintegrate…
- If cottheta = 4/3 , then the value of costheta -sintegrate heta /costheta…
- If cosecA = 4/3 and A + B = 90, then secB is ………..A. 3/4 B. 1/3 C. 4/3 D. 7/3…
- If θ is the measure of an acute angle and √3 sinθ = cosθ, then θ is ____A. 30…
- If tana = 5/12 then the value of (sinA + cosA) secA is …..A. 12/5 B. 7/12 C.…
- If tantheta = 4/3 then the value of root 1-sintegrate heta /1+sintegrate heta…
- In ΔABC, if m∠ABC = 90, m∠ACB = 45 and AC = 6, then area of ΔABC is …..A. 18…
- If cos^2 45 - cos^2 30 = x ⋅ cos45 ⋅ sin45, then x is …….A. 2 B. 3/2 C. - 1/2…
- If A and B are complementary angles, then sinA ⋅ secB is ____A. 1 B. 0 C. —1…
- The value of tan20 tan25 tan45 tan65 tan70 is ………….A. —1 B. 1 C. 0 D. √3…
- If 7θ and 2θ are measure of acute angles such that sin7θ = cos2θ then 2sin3θ —…
- If A + B = 90, then cotacotb+cotatanb/sina secb - sin^2b/cos^2b is ….A. cot^2…
- For ΔABC, sin (b+c/2) = ……..A. sin a/2 B. sinA C. cos a/2 D. cosA…
- sin^4theta -cos^4theta /sin^2theta -cos^2theta = ………..A. 1 B. 2 C. 3 D. 0…
- If 7cos^2 θ + 3sin^2 θ = 4, then cotθ is ___A. 7 B. 7/3 C. root 3 D. 1/root 3…
- If tan5θ ⋅ tan4θ = 1, θ is ____A. 7 B. 3 C. 10 D. 9
- If A and B are measures of acute angles and tanA = 1/root 3 and sinB = 1/2 ,…
- In ΔABC, m∠A = 90. If AB = 5, AC = 12 and BC = 13, find sinC, cosC, tanB, cosB,…
- In ΔABC, m∠B = 90. If BC = 3 and AC = 5, find all the six trigonometric ratios…
- If cosA = 4/5 find sinA and tanA.
- If cosec θ = 13/5 find tan θ and cos θ.
- If cosB = 1/3 , find the other five trigonometric ratios.
- In ΔABC, m∠A = 90 and if AB : BC = 1 : 2 find sinB, cosC, tanC.
- If tan θ = 4/3 , find the value of 5sintegrate heta +2costheta /3sintegrate heta…
- If sec θ = 13/5 find the value of 5sintegrate heta +3costheta /5costheta…
- If sinB = 1/2 prove that 3cosB - 4cos^3 B = 0.
- If tanA = √3, verify that (1) sin^2 A + cos^2 A = 1 (2) sec^2 A - tan^2 A = 1…
- If cos θ = 2 root 2/3 , verify that tan^2 θ - sin^2 θ = tan^2 θ⋅ sin^2 θ…
- In ΔABC, m∠B = 90, AC + BC = 25 and AB = 5, determine the value of sinA, cosA…
- In ΔABC, m∠C = 90 and m∠A = m∠B, (1) Is cosA = cosB? (2) Is tanA = tanB? (3)…
- If 3cotA = 4, examine whether 1-tan^2a/1+tan^2a cos^2 A - sin^2 A.…
- If pcot θ = q, examine whether psi ntheta -qcostheta /psi ntheta +qcostheta =…
- State whether the following are true or false. Justify your answer: (1) sin θ =…
- cos60 = 1 - 2sin^2 30 = 2cos^2 30 - 1 = cos^2 30 - sin^2 30 Verify:…
- sin60 = 2sin30 cos30 Verify:
- sin60 = 2tan30/1+tan^230 Verify:
- cos60 = 1-tan^230/1+tan^230 Verify:
- cos90 = 4cos^3 30 - 3cos30 Verify:
- sin30+tan45-cosec60/sec30+cos60+cot45 Evaluate:
- 5cos^260+4sec^230-tan^245/sin^230+cos^230 Evaluate:
- 2sin^2 30 cot30 - 3cos^2 60 sec^2 30 Evaluate:
- 3cos^2 30 + sec^2 30 + 2cos0 + 3sin90 - tan^2 60 Evaluate:
- In ΔABC, m∠B = 90, find the measure of the parts of the triangle other than the…
- In a rectangle ABCD, AB = 20, m∠BAC = 60, calculate the length of side bar bc…
- If θ is measure of an acute angle and cosθ = sinθ, find the value of 2tan^2 θ +…
- If α is measure of acute angle and 3sinα = 2cosα, prove that (1-tan^2alpha…
- sin(A + B) = sinA cosB + cosA sinB, If A = 30 and B = 60, verify that…
- cos(A + B) = cosA cosB - sinA sinB If A = 30 and B = 60, verify that…
- If sin(A - B) = sinA cosB - cosA sinB and cos(A - B) = cosA cosB + sinA sinB,…
- State whether the following are true or false. Justify your answer: (1) The…
- cos18/sin72 Evaluate:
- tan48 — cot42 Evaluate:
- cosec32 — sec58 Evaluate:
- cos70/sin20 + cos59 ⋅ cosec31 Evaluate:
- sec70 sin20 — cos20 cosec70 Evaluate:
- cos(40— theta) — sin(50 + theta) + cos^240+cos^250/sin^240+sin^250 Evaluate:…
- cos70/sin20 + cos55cosec35/tan5tan25tan45tan65tan85 Evaluate:
- cot12 ⋅ cot38 ⋅ cot52 ⋅ cot60 ⋅ cot78 Evaluate:
- sin18/cos72 + √3 (tan10 tan30 tcm40 tan50 tan80- Evaluate:
- cos70/sin20 + sin22/cos68 - cos38cosec52/tan18tan35tan60tan72tan55 Evaluate:…
- sin48 sec42 + cos48 cosec42 = 2 Prove the following:
- sin70/cos20 + cos6c^20/sec70 — 2cos70 cosec20 = 0 Prove the following:…
- Prove the following:
- cos (90-a) sin (90-a)/tan (90 - phi) = sin^2a Prove the following:…
- Express the following in terms of trigonometric ratios of angles having measure…
- For ΔABC, prove that (1) tan (a+c/2) = cot b/2 , (2) cos (b+c/2) = sin (a/2)…
- If A + B = 90, prove that root tanatanb+tanacotb/sinasecb = seca
- If 3 θ is the measure of an acute angle and sin30 = cos(θ — 26), then find the…
- If 0 θ 90, θ, sinθ = cos30, then obtain the value of 2tan^2 θ — 1.…
- If tanA = cotB, prove that A + B = 90, where A and B are measures of acute…
- If sec2A = cosec(A — 42), where 2A is the measure of an acute angle, find the…
- If 0 θ 90 and secθ = cosec60, find the value of 2cos^2 θ — 1.
- cos^2theta + 1/1+cot^2theta = 1 Prove the following by using trigonometric…
- 2sin^2 θ + 4sec^2 θ + 5cot^2 θ + 2cos^2 θ — 4tan^2 θ — 5cosec^2 θ = 1 Prove the…
- 1/1+costheta + 1/1-costheta = 2costheta c^2theta Prove the following by using…
- Prove the following by using trigonometric identities:
- root 1-sintegrate heta /1+sintegrate heta = sectheta -tantheta Prove the…
- sectheta +tantheta /costheta c theta +cottheta = costheta c theta -cottheta…
- cottheta +cosectheta -1/cottheta -cosectheta +1 = cosectheta +cottheta Prove the…
- (sinθ + cosecθ)^2 + (cosθ + secθ)^2 = 7 + tan^2 θ + cot^2 θ. Prove the following…
- 2sec^2 θ — sec^4 θ — 2cosec^2 θ + cosec^4 θ = cot^4 θ — tan^4 θ. Prove the…
- (sinθ — secθ)^2 + (cosθ — cosecθ)^2 = (1 — secθ ⋅ cosecθ)^2 . Prove the…
- Prove the following by using trigonometric identities:
- tantheta -cottheta /sintegrate heta costheta = sec^2theta -costheta c^2theta =…
- sectheta -tantheta /sectheta +tantheta = 1-2sectheta tantheta +tan^2theta Prove…
- root sec^2theta +costheta c^3theta = tantheta +cottheta Prove the following by…
- Prove the following by using trigonometric identities:
- tantheta /1-cottheta + cottheta /1-tantheta = 1+tantheta +cottheta = 1+sectheta…
- sin^4 θ - cos^4 θ = sin^2 θ - cos^2 θ = 2sin^2 θ - 1 = 1 - 2 cos^2 θ. Prove the…
- tan^2 A — tan^2 B = Prove the following by using trigonometric identities:…
- 2(sin^6 θ + cos^6 θ) — 3(sin^4 θ + cos^4 θ) + 1 = 0 Prove the following by…
- If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p^2 — 1) = 2p.…
- If tanθ + sin = a and tanθ — sinθ = b, then prove that a^2 — b^2 = 4 root ab…
- acosθ + bsinθ = p and asinθ— bcosθ = q, then prove that a^2 + b^2 = p^2 + q^2 .…
- secθ + tanθ = p, then obtain the values of secθ, tanθ and sinθ in terms of p.…
- sec38/cosec52 + 2/root 3 ⋅ tan17 tan38 tan60 tan52 tan73 — 3(sin^2 32 + sin^2…
- - cottan (90 - theta) + cos6ctheta sectheta (90 - theta)
- If sinA + cosA = √2 sin(90—A), then obtain the value of cotA.
- If cosecθ = √2, then find the value of 2sin^2theta +3cot^2theta /4tan^2theta…
- If tantheta = 8/15 then evaluate (1+sintegrate heta) (2-2sintegrate…
- If costheta = b/root a^2 + b^2 , 0 θ 90, find the value of sinθ and tanθ.…
- If θ is the measure of an acute angle such that bsinθ = acosθ, then…
- Which of the following is correct for some 0 such that 0 ≤ θ 90?A. 1/sectheta…
- If tantheta = 1/root 5 , then cosec^2theta -sec^2theta /cosec^2theta…
- If tan^2theta = 8/7 , then the value of (1+sintegrate heta) (1-sintegrate…
- If cottheta = 4/3 , then the value of costheta -sintegrate heta /costheta…
- If cosecA = 4/3 and A + B = 90, then secB is ………..A. 3/4 B. 1/3 C. 4/3 D. 7/3…
- If θ is the measure of an acute angle and √3 sinθ = cosθ, then θ is ____A. 30…
- If tana = 5/12 then the value of (sinA + cosA) secA is …..A. 12/5 B. 7/12 C.…
- If tantheta = 4/3 then the value of root 1-sintegrate heta /1+sintegrate heta…
- In ΔABC, if m∠ABC = 90, m∠ACB = 45 and AC = 6, then area of ΔABC is …..A. 18…
- If cos^2 45 - cos^2 30 = x ⋅ cos45 ⋅ sin45, then x is …….A. 2 B. 3/2 C. - 1/2…
- If A and B are complementary angles, then sinA ⋅ secB is ____A. 1 B. 0 C. —1…
- The value of tan20 tan25 tan45 tan65 tan70 is ………….A. —1 B. 1 C. 0 D. √3…
- If 7θ and 2θ are measure of acute angles such that sin7θ = cos2θ then 2sin3θ —…
- If A + B = 90, then cotacotb+cotatanb/sina secb - sin^2b/cos^2b is ….A. cot^2…
- For ΔABC, sin (b+c/2) = ……..A. sin a/2 B. sinA C. cos a/2 D. cosA…
- sin^4theta -cos^4theta /sin^2theta -cos^2theta = ………..A. 1 B. 2 C. 3 D. 0…
- If 7cos^2 θ + 3sin^2 θ = 4, then cotθ is ___A. 7 B. 7/3 C. root 3 D. 1/root 3…
- If tan5θ ⋅ tan4θ = 1, θ is ____A. 7 B. 3 C. 10 D. 9
- If A and B are measures of acute angles and tanA = 1/root 3 and sinB = 1/2 ,…
Exercise 9.1
Question 1.In ΔABC, m∠A = 90. If AB = 5, AC = 12 and BC = 13, find sinC, cosC, tanB, cosB, sinB.
Answer:![](data:image/jpeg;base64,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)
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 2.In ΔABC, m∠B = 90. If BC = 3 and AC = 5, find all the six trigonometric ratios of ∠A.
Answer:Pythagoras theorem
32 + AB2 = 52
⇒ AB = 4
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCACEAIEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aqP9r2f9sf2Tvb7T5XmY2/Lj0z645x6VckkWKNpHYKiAsxPYCuIMWsNpp19baDH2n+0QN7ed5eMbMYx/q+MZralTU73/AKZlUm42sdzRTIZUnhSaJtySKGU+oPIp9YmoUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFcz4517UtA0uCfToUJkl2yTOhdYhjjgep4z/wDWrpqzPEXGhXP0X/0IVpTtzq6uRUvyOwvh7UpNX0G0vp41immjzIi9FbuPUfQ8itKsWeKaxunvrFC+85uLYf8ALX/aX0f+fT0Nalpdw3tslxbuHjfofT1BHY+1E49VsEZdHuTUUUVmWFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFZniL/kBXP0X/0IVp1meIv+QFc/Rf8A0IVdP416kT+Fk7feP1qhLFNY3LX1ihcvzcWw/wCWv+0vo/8APoexq+33j9aSrTsS1cntLuG9tkuLdw8b9D6eoI7H2qasWWGayuWvrFN5fm4th0m/2l9H/n0PY1qWl3De2yXFu++N+h7j1BHY+1RKNtVsVGV9HuTUUUVBYUUUUAFFFFABRRRQAUUUUAFFFFABWZ4i/wCQFc/Rf/QhWnWZ4i/5AVz9F/8AQhV0/jXqRP4WTt94/WkpW+8frSVQgqlNDNZXLX1gm8vzcWw4E3+0vo/8+h7GrtFNOwmrk9pdw31slxbvvjfoehB7gjsR6VNWLNDPZXLX9gm8tzcW44Ew/vD0cfr0PY1qWl3DfWyXFu++Nuh6EHuCOxHpUSjbVbFRlfR7k1FFFQWFFFFABRRRQAUUUUAFFFFABWZ4i/5AVz9F/wDQhWnWZ4i/5AVz9F/9CFXT+NepE/hZO33j9aSlb7x+tJVCCiiigAqlNDNZXLX9gm9m5uLcHAmH94ejj9eh7GrtFNOwmrk9pdw31slxbvvjb8CD3BHYj0qasWaGezuWv7BN7NzcW4OBMPUejj9eh7GtO0u4L62W4t33xt+BB7gjsR6VEo21WxUZX0e5PRRRUFhRRRQAUUUUAFFFFABWZ4i/5AVz9F/9CFadZniL/kBXP0X/ANCFXT+NepE/hZO33j9aSlb7x+tJVCCiiigAooooAKpTQz2dy1/YJvZubi3BwJh6j0cfr0PY1dopp2E1cmtLuC+tluLd98bfgQe4I7EelT1izQz2dy1/YLvZv+Pi3BwJh6j0cfr0PY1p2l3BfWy3Fu+5G9sEHuCOxHpUSjbVbFRlfR7k9FFFQWFFFFABRRRQAVmeIv8AkBXP0X/0IVp1ieLb+0sNAma6nSESFVTd1Y5BwB1PANXT+NepE/gZfb7x+tJTYpo7qJLi3kWWKUbkdDkMD0INOwfQ1QgoowfQ0YPoaACijB9DRg+hoAKKMH0NGD6GgAqlNDPZ3LX9gu5m/wCPi3zgTD1Ho4/Xoe1XcH0NGD6GmnYTVya0u4L62W4t33I3tgg9wR2I9KnrFmguLO5a/sELO3+vt+gmHqPRx69+h7Gp9M8R6RrE729jfRyzxrueLkMvY8H0PB9KmUOsdioy6Pc06KYs0TSNEsiGReqhhkfhT6zLCiiigDK8SX02m6JLdW8qQyK8a+Y65ChnUE4+hNczf6fH4tvoNNk1uK/SJGmFxaxqrW54GG5IIYE+4K11OvWE+paS9tbeX5pkjdfMJCna6tg4B9Kqw6bqNzrNrf3sdnbLaK4VbdmdpCwxgkgYA6455xXVSlGML9df+B5nPUi5St00/wCCPsvCejWNlDapaB1iXbucks3qT7mp/wDhHtI/58Y/1rSorD2k+7NeSPYzf+Ee0j/nxj/Wj/hHtI/58Y/1rSoo9pPuw5I9jN/4R7SP+fGP9aP+Ee0j/nxj/WtKij2k+7Dkj2M3/hHtI/58Y/1o/wCEe0j/AJ8Y/wBa0qKPaT7sOSPYzf8AhHtI/wCfGP8AWj/hHtI/58Y/1rSoo9pPuw5I9jN/4R7SP+fGP9a5i58H2vha01PWdOuLjzlgdYVYjFurEbiOMnA6Z6Yruaa6LIjI6hkYYZSMgj0qo1pp6vQUqcWtFqY0uh6FZ2tvMI4rT7O6Ml0hCuTkAZfq27ODnrmtusmDw1p0EsTATukDboYZJ3eOIjoVUnHHb07VrUqkr9W/UcFbpYKKKKzLCiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAP//Z)
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIIAAAAqCAMAAABIknNMAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZmZmZpCQZpC2Zra2ZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNvbkNv/tmYAtmY6tmZmtpA6ttv/tv/btv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bwzW38gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACb0lEQVRYR+1YyVLDMAyNA6WBAg17CVtdttA2KfH/fxxanMQcwMrYhWEGHZoyKNKzJOtJTZJ/iRYBU+ytwZgpFEp6hH8IpEznAi2ZSqXYWJOPAciTgk+B1FlECPoim6JPU6DzJqeY+KTJz+NBaE5f2kwwBFEU9LSKB6EcJ+XOaxuF99sRfvdJNVnHg2CK66TOrrt63Hv2ucdQncyTeBDqydpWAdWC0YL4mgKKByCUiDxcSrqKmAl5OVb0DkgUCBhSOBEaYwimoMLwS7REVHQFuSUMuZSUCD9OgQa7BQhwdMahFTUJv0Tqjk2uFIQBHzsv3KB3Z37vqAH1ECkMMn9/Vsvy3y/gr/dtnbb89/MYuqvS8t/WIdRnUJjQVvA5ge6+aFtdx3/bhlBns2STI8/Ak9paG4WO/7YNQdvxQmODIb6zEHr+6yDotrEPuc/el5qcOxk/6dNC6Pnv+yh0sD598UWuV7auWwjUbC2Env985gL//2UUHP7bciIsyQDHYU04teDwX+Apva+XapaYRxiigN42Fsc0eXxw+M9rI1ThLVOjezCyzJTiHWSh0iuH/2TjRyiMkPc/r1ArbHSHgjXCLA+UGlkqCPHP97lfoUp1vDZPkilAp/eQeImmAJ8zrdUZzY43gsNRKyzjTK/dwAg2h05iWjjn+gJBUaAVSjw6s8nN3aFk6fH5x7jTxIgrlHCfZZs4bUrqVoCAE0Er1CAIYHqVxUwEuR+YCJyhheO+JxTOCiXZJl1rNf8sESzOpeRTSS4lrbW8BUYQZ4UyRQqMI2lNoDm3nBQOwTZoXqHMAphm91LQoDd3HSeFY/j7Fj4AQ/g9bszUEG0AAAAASUVORK5CYII=)
Question 3.If cosA =
find sinA and tanA.
Answer:
![](data:image/jpeg;base64,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)
Let AB = 4 and AC = 5
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE4AAAAqCAMAAAAqEZ1jAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZpCQZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tmZmttvbttv/tv/btv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bMPEXCwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABaklEQVRIS+WV607DMAyFbRgL62hgwNrCwrWDNk3e//mwvRYVCVoiGQlp/rFUXfMl59hxAI4j6pOdolBvNHHB3mjiXN4q4tqsUcSFyx3o4WKZA+PqQiW3Lfahg5M96Yn9C5zuqSD/FAtPJaWpkFieNalzJr5vdQ1xt4YKPin8hpJC9cxj9kxTu4peXLDIcPWSqtabLXS2ABnL01dgv+KdeFYvoaY3KeF6s92SZnlTQLCkz2eEiyWvMj6IbjidP3kqk1mXjPL7gKsnecfMWPIy0/G5BvaYAXeY/HaOC5ZYy3dJar/ZHYHeDWG5KVKnwJHaWbGDmEPBslNiGzvaiqvBzqsdeVHjFuIjNWfMKbNEkBRvloNrwSaphb3BxT07ZhC52rqKHtZNsIgE5+GXvL3Wve1pA4oN6mtxztXV7P9HhltReWglFmLF3Uj36vHJnXYyxYmncYLluLnp7c6RbdLJdaK7pjO25nvmX8cHH7kcnQUJm2MAAAAASUVORK5CYII=)
BC = 3 (by Pythagoras theorem)
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAgCAMAAABpVqV8AAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpCcOpDbZgAAZgA6ZgBmZjoAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttvbtv/btv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bq7fipAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABQklEQVRIS92VfVPCMAzGk/lCBUXmG2zqhjJe1rXf/+uZhA2lcq5Czj/M3Xrckf36PEnaAfzLWA4xedJw5vNXWCWFBooY9kqJ1LxMdSSVeK5EAlibmYIod1eA3SPVGHJ9drmJ2Gppgin4TqpRq7nlo5lEaNqlNDliMt7USO54IbFj+dPdv8fZa1FcDP9MBZGqWTMqoNpWrBpAdbaIF+VScmCvdyTBsSmfzYKeQIldHCzgHEdvvHGriUgCF7rPBr2aWjjnrYZ4QSZCUiUpv7FHrLWhrQMSDxxQC74O2c/u2ASUnxXv3NUylS7tt9f5t2YKzQPly/aycMXbCrk03l6TG8Qbbh3NkiwuRbylh0Txz3hUb3P+LoGbrHNAK427SYwrksjd9swrxP6deQrQZ3xSTw2fh/f4sUQ/pyFQ+0odq+Lwex+f1hmGKVPXSgAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
Question 4.If cosec θ =
find tan θ and cos θ.
Answer:![](data:image/jpeg;base64,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)
Let ∠A be θ
BC = 5k, AC = 13k
⇒ AB = 12k (by Pythagoras theorem)
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK8AAAArCAMAAAA5bSiJAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kJxmkLyQkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv/btv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///bC3ns5wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADKElEQVRYR+1YiXLaMBCVTClOSxsSkqZxjjYFchQTWmzj//+z7tuVZRxsiKmdERk0A2Nb0upp9fbQKnVo71gDkdZB1fZC7Y2c23oEvPy33iIH8cb+HuKtOPWW9Btp79eDrwdYNT7T3nccsA5mvvejqscO1l9Jv4bENFd7AUkbqdhnmoAP6fMnjeFGZBOEjnT/ST3rU4LrX6o5Tjj2+5fzo1F5z0iFgIPB6Y08mddkGKgQKIXW9JRe9RbpGJ9EZDN4SXh69XGhxvSjFSCc0JtlX/YwNurGYAHGePkVcwp4+ZPZD4tsoIl503rLIURiYV4hM/z1HgyUwQZKYF434q30fDX3IHjDznQ51GidaRHvek8NvOn83GcnbVRQE1upnyzq157gmn5tz+vxgr/5ITQANmdpT6ibM87yd60HeGWw5a+dW+Qvv1nSNIWXGDsTMyGTn7FDM/wt72F7C707lZzb+BZi7j182WCRf8bI5NuGoLLDFiL92dcf7jBz5mvvAgeo4d4IdllPAJ6TgifaG/whT4vRtD2a24UQ+nyBz+RxSQcT8r5/hzKoIQdREf6tf9hBBTtNSW9pb+IUNzZH8MI0k2FmQRsQvy3e+KgiyxRD3Z7TGfqVbKi6Z9uRbdJOFSIOkM156f+AuDqV7A8RqZgGISEShyhR06UmWdtqGiT5k+DNQ4ArmHOG5mEEXtokA3rV4MacH6Btp3Vb+yvDK/H9lfq1e3iLh8wDFNOgPL47zN+VtIJRsn9YUsafN2f48CINYrxyMaFk1q0G27r/WUyDOH8iAlN8c82dcT4UFNIgkz+phC67Jw0pN8StY58aXYr2CS6udwe8LWog7Py+5dqPrdio5JriJBJsUxVqcfX6okPv+CnLS6RigzpKesPFClMVqi+1vRnszSXGU0Oqwk4+/rJSFWpv9fqSxd6yohOnVhPdf7Qpa9ZTX3I7M3K8NlVRVHrsTrNE0DH/YfHmFRtSzNzvOXglwIkZ/p7KjRB8AHVBEPeuBIxXUyH3gWvRpmIDt5CckQHaqlA7TNxNanhC1YwuruJZxSa5ppLNMeobpiq0m+DDrCY18A8RnnT+pmAergAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIIAAAAhCAMAAAAiVYOPAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tmZmttu2ttvbttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bMH7hsQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACIklEQVRYR+1Wa1PCMBDMVUErivKUhwKtrwLaps3//2/eXZrCOEJSTeWLN0Pboclls9m7rhD/4YuBYniJqWQIGK1XX1lr5cmuz95wQobXnNFYQsU86GMM0LONdXuvHp6H0xKCiNqpbdb6dkwQitFSbIKVbbTT+6wnIspJLMgRgbFFYqiSV34gxCteXWSkhYGVBMRXQUgGNrRO77UMcfOshZmDFioIG09SoJ0oWpmpSOxaqFhIrAjUAhVr5TVnKcaoK2cWVHRBaVFDIjsqHdxZaq8xNQNaHaD9zgdyZ8UsEtaMKIblAR4+7owKhi8nC65wGboU2S8xSuxRAa5D9y5113yOf/RSVhiy5adsjoGU4QS76hS3i/cZalrN2ql6xB9D0NdmwzRU7m9EO+9bdlOtFoA9CJH+B8NRIU4TDNH6ztcYbl742Y2FCpZ5sFH2ZUJ11hqCpn3TgRYdyB9p4RsWEMQ2xOX5iApuPGU48bpPgtMEIzcSodYCykAvn3BfYCfQbCQwEeqJ+tsAKwJX5tKgDzqKAbtj8zUpxDqE1pIUgP2Vvgj5HB+4zeZjCPrNEuAtuy4exgzBxFvaOolK7yjDfqoih5MrveO6A8F9nXUOjzXe0cE2cpLSO6q5d+9Y43PSlHf8AYR84Uc5xjvWhxDBuR8IxjvqLufkiauD2HqxIzvvSF3OqZdp71iMVn4c0c47ptTlXKgtvSMN91WUfkr7hFk+AToUNfph+TpKAAAAAElFTkSuQmCC)
Question 5.If cosB =
, find the other five trigonometric ratios.
Answer:
![](data:image/jpeg;base64,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)
Let BC = k and AB = 3k
⇒ AC =
(Pythagoras theorem)
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 6.In ΔABC, m∠A = 90 and if AB : BC = 1 : 2 find sinB, cosC, tanC.
Answer:![](data:image/jpeg;base64,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)
![](data:image/png;base64,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)
AB = k and BC = 2k
AC =
(by Pythagoras theorem)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.If tan θ =
, find the value of ![](data:image/png;base64,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)
Answer:Let’s divide both numerator and denominator by cosθ, we get
![](data:image/png;base64,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)
put tanθ =
in this equation.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 8.If sec θ =
find the value of ![](data:image/png;base64,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)
Answer:![](data:image/jpeg;base64,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)
Let ∠B = θ
![](data:image/png;base64,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)
⇒ AB = 5k, BC = 13k
⇒ AC = 12k (by Pythagoras theorem)
we know that
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAuCAMAAABtVWkHAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kJxmkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///btr0w2wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAECElEQVRoQ+1ZDVfTQBC8C6JBQasUFGlBkYJKS7Vpmv//z9zZva+EJlzSBovPe4++5LK5zO3t7c4NSv1v/6QHMq1HdROb6uR6+5Muxhrt8GeXoTOg5Z+HLesDrVoNXyk1T/fuOsBdpk+NthgTWjWtX9OGWTDamtaPbwVtpo+VWp7o5DOu9WiWJl918u17qgdAU3riuvRbQmuClyx0MsoQrcuUAwSRUNy/1jA3Q3ZYvsor3rfL9JxCgj60TA/P5wfXGaL5nqcRPLmWdUBXcSlX5jYfjtQUGCWc6YrGXhQTdMmQW0I7T18u1IT+GPsyJYTmo8W4/ISR0WMYCyxGy7d4p4SWu8xseMiNGyeFZLCg7YYB8VkTjrLZqaPyBLfSZYCMzG0j2voIbzMFiQRqqyHnMkoOJbTTvbvKkxZoi/lpykm5aT92RGsXa71v3arGo0Xc+gVog6rO1vnWXZR8i97KE6C1mcS4zVmU45bvXLhsAy1XB9khmrb5jFMYB1mmKYZnsm3CJ7zLpsmVyk9dLZvC4hbZa7Dw3bDMPzWUkLYTwCYz+1nNUp2cYek00hZhfJPqF1cYMXgyQhSTc29oa/6izAprmhxZ7MOUus/QTRmW5nlD2fb3UIy2kxRq51dDANr6I8q++ELzkiTYsfWJtjI2tmPuQrIT3idEK5tzE9ZmArLTRNu+xKVwWzm57cfr7fMLLpxMhYTqgPRIApT6uEMNrKO4tMWdqI5wJEHr0/qOIGb3LY/CAoecbEq7tpkfaCfMA9A2CefN5n2jD39gBF/gpI5H+tbN4CkuCCbR8306VJXR7mjcwq3zlCm0JWYcHJwTVsTmfduBSEDIegrNkBmtHDk6nWQ3i8zGt3E+yk/It/5szxyJApdq2Y4lMJVfEL15J8RJ+JBwJJqCTj706Ka/OjSYI6/QX0xvLRxgWDnHe3PsHTxm0eKrXU1N3XucCD1u0RVCi/c8WjlO0B6oyjX6iLQ9Og7IwTgUdkSPcTqN4S5O42kBI9I08C0rHkw3y3JNPkZGFN+ukW+cThNwF6P+RGKINzPSKG8y0hDox6ZFkWtQyUVTZLSBsGPlG3feddzFGcXDiLQM4xaoiguraHm5hmEwqDXyTaAqGe7ijSIxxJuFaHGdOaELrpYPs41BWxJ2/DTkeCTcxWs88TAiLUs5gfBN8NWKFFb1LYb2xd4rH9QN7tLjYaCEdjV8/xFiOaMVucadnESYterZOrSWu/R4GChXh4mIFaFcQ7KLgXysbkncfSDfOJ0m4C7GKHJ9o80qldewt0CuuScOIvoEyTEizlhhx1ITq9NY7uKMolF0NMzkiNSnpNARWfU1yl2cvp4J2vHg0pw+n4NvxyDMkp0a/tu4pZXc3WH+AEjKm1cS5DxbAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
putting the above values in the given equations.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Question 9.If sinB =
prove that 3cosB – 4cos3B = 0.
Answer:Let AC = 1 and AB = 2.
![](data:image/jpeg;base64,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)
⇒ ![](data:image/png;base64,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)
∴ BC = √3 (by Pythagoras theorem) we know that
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAuCAMAAAABUzsCAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kJxmkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv/btv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///brTZjugAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADDUlEQVRYR+1Yf3PSQBC9jQJRabGk1dpAqyW2YkPR/CDf/5v5dvdyBAcLjA2X6XT/gGQIcy97b9++W2Neo8MZSOnNQ4fhGZO+4vuv/el+/n7eUPAF71g9vica4aKcEgWjzJjiXH/xGWlwOjcpxaaaDLIqCWa46GfVdT8zRXhllmHsEx3qA4jMKhooigJwVtEYFyeZSYARqD3jk/plKDU+852GP3AjON0vvlBqfUiqlhchYaONARF7D8BHHJ71x+Fj/sn+cizDgc2fr7zV61r+jU3ORGR8TD1OqHfqCcSUULb3wFaEY1N+Znwo2/IcZZHTlakWngUmPbsh6iF1KAsa/Y4oLqeg4SnXyyKk4FIL52VGhXdnoe9ocMmVtbR2EKOUnHx0M0T1a8nyBxE2ggLoJn+fzIHDGgwVKG1DHkP0KLK6NEG7qQ2G4vMuo3XvThhN02DYBkkNh5FIy+Q4Givr/dNv+XQGY7/8OcwtXKz5pfh0N9VgdIN/W/IHhGww1JutQE0XHva35j9XxabBUO+bezZocBiwEXcz2AkYDAbpDAZsOfqHZ3kRG9H7xqyDr+BuuzYYJQ5YZx6l73mWribEGpQfUXQOA26PZ0LXJ6N4t+uJwxbe82lbYrudxO4n9lzxsMfW+Hir+1kBX/yVgtv7UOYFrrHDQct5zc4NQIgY3McjObcj/AtK1sZwoZE/VSnOU07DuXmEJqgecGO3Xs3NDYpwOJPZglUymTK0MFzgrNU9Wc6TqZQL1hW4rrFbfG5uINaNz036A/+jleFCk3+8ZjW1q0kLajR2gbGeGwg+uXX4WhkuNPHxdS57LKMCpLPR2C2+em6wDV8bw4WN+gWihJEpvn/mT+pmS/5s739WV7yBbxV9/MQjF8u/gXJQsUj+1oZ3C75Whgub+pxwzTI+wFoo721jl1q4m7m5gbyC1EcRjrLyArdtDBf+6m/W8OT0IaS33NddY2fnywcdOzeA4FHMJl0scXD5CwJ4hOFCrucB5V/HAsoi4tJZfJPRtT1OdTN/E51MMeN1VvrS4w/DGG3cXLApqQAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
= 0
which is equal to the R.H.S
Question 10.If tanA = √3, verify that
(1) sin2A + cos2A = 1
(2) sec2A – tan2A = 1
(3) 1 + cot2A = cosec2A
Answer:![](data:image/jpeg;base64,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)
Let BC = √3 and AC = 1
⇒ AB = 2 (by Pythagoras theorem)
we know that
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAuCAMAAABtVWkHAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kJxmkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///btr0w2wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAECElEQVRoQ+1ZDVfTQBC8C6JBQasUFGlBkYJKS7Vpmv//z9zZva+EJlzSBovPe4++5LK5zO3t7c4NSv1v/6QHMq1HdROb6uR6+5Muxhrt8GeXoTOg5Z+HLesDrVoNXyk1T/fuOsBdpk+NthgTWjWtX9OGWTDamtaPbwVtpo+VWp7o5DOu9WiWJl918u17qgdAU3riuvRbQmuClyx0MsoQrcuUAwSRUNy/1jA3Q3ZYvsor3rfL9JxCgj60TA/P5wfXGaL5nqcRPLmWdUBXcSlX5jYfjtQUGCWc6YrGXhQTdMmQW0I7T18u1IT+GPsyJYTmo8W4/ISR0WMYCyxGy7d4p4SWu8xseMiNGyeFZLCg7YYB8VkTjrLZqaPyBLfSZYCMzG0j2voIbzMFiQRqqyHnMkoOJbTTvbvKkxZoi/lpykm5aT92RGsXa71v3arGo0Xc+gVog6rO1vnWXZR8i97KE6C1mcS4zVmU45bvXLhsAy1XB9khmrb5jFMYB1mmKYZnsm3CJ7zLpsmVyk9dLZvC4hbZa7Dw3bDMPzWUkLYTwCYz+1nNUp2cYek00hZhfJPqF1cYMXgyQhSTc29oa/6izAprmhxZ7MOUus/QTRmW5nlD2fb3UIy2kxRq51dDANr6I8q++ELzkiTYsfWJtjI2tmPuQrIT3idEK5tzE9ZmArLTRNu+xKVwWzm57cfr7fMLLpxMhYTqgPRIApT6uEMNrKO4tMWdqI5wJEHr0/qOIGb3LY/CAoecbEq7tpkfaCfMA9A2CefN5n2jD39gBF/gpI5H+tbN4CkuCCbR8306VJXR7mjcwq3zlCm0JWYcHJwTVsTmfduBSEDIegrNkBmtHDk6nWQ3i8zGt3E+yk/It/5szxyJApdq2Y4lMJVfEL15J8RJ+JBwJJqCTj706Ka/OjSYI6/QX0xvLRxgWDnHe3PsHTxm0eKrXU1N3XucCD1u0RVCi/c8WjlO0B6oyjX6iLQ9Og7IwTgUdkSPcTqN4S5O42kBI9I08C0rHkw3y3JNPkZGFN+ukW+cThNwF6P+RGKINzPSKG8y0hDox6ZFkWtQyUVTZLSBsGPlG3feddzFGcXDiLQM4xaoiguraHm5hmEwqDXyTaAqGe7ijSIxxJuFaHGdOaELrpYPs41BWxJ2/DTkeCTcxWs88TAiLUs5gfBN8NWKFFb1LYb2xd4rH9QN7tLjYaCEdjV8/xFiOaMVucadnESYterZOrSWu/R4GChXh4mIFaFcQ7KLgXysbkncfSDfOJ0m4C7GKHJ9o80qldewt0CuuScOIvoEyTEizlhhx1ITq9NY7uKMolF0NMzkiNSnpNARWfU1yl2cvp4J2vHg0pw+n4NvxyDMkp0a/tu4pZXc3WH+AEjKm1cS5DxbAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
(1) sin2A + cos2A = 1
⇒![](data:image/png;base64,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)
(2) sec2A – tan2A = 1
⇒![](data:image/png;base64,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)
(3) 1 + cot2A = cosec2A
⇒ L.H.S
⇒ ![](data:image/png;base64,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)
R.H.S
⇒ ![](data:image/png;base64,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)
L.H.S = R.H.S
Question 11.If cos θ =
, verify that tan2θ – sin2θ = tan2θ⋅ sin2θ
Answer:![](data:image/jpeg;base64,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)
Let ∠B = θ
and BC = 2√2 and AB = 3
⇒ AC = 1 (by Pythagoras theorem)
we know that
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAuCAMAAABtVWkHAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kJxmkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///btr0w2wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADzklEQVRoQ+1ZDXfSQBDMpbZNlSpaWq2FohWolgBKCPn//8yd3ftIAiFEEgw+7z19JLd3mdvb252bet7/9k96IFKqX7SwUPmjli06Alr+b7NFrUO7Ck4ObcF+N+LbSPnfvgeqi2+ubpX/GVur+rPA/1rUY43VW/KtDl4aq/w+zTbyVgEHCCIhmb9WMNdTHh7Ikeq8eHN1Q2CDB2+BvV0FnYfF1Wh7z8gLAQbGyVB+6ce41/dCYJRwpl/J4HKZjPFKpqwDLU2dDC6W3pj+0fyYmrDrj+Z7GBl1w1hgMVp+xJgMWn6lV8NTHtzkSNPX1j1MiM/y/Oawb/bAUIw1kL5+3Im2MNdVWoGgDc+m655CO5tm0W72VECbLO4CTsraAZWQbc2LWd/avdvwre3ZHy3i1m3AwVBddF5KyLpIs3G70QO0Ymzj1o7Nxi0/2XCpBy1F6kwOBx3zGacwHbfbe/iUhf6TF9/ZWhZi7DOyV3fpXsMy/rSjhFReQKTeBOrVE8bNAuXfY+sUEhqB3tbTR3yTcyfK7/6kzAprWhyNPcck9PoerynDkgcmlG1/9cSolqRQUOZtTqi8/D8akHyhdUkS3Nn+DtrcV3Ec4545NzvwtgKtHM5y1qbDbstyinvKtqt6P5fCunJy9c8XjYgf6fR1l0yFhOqA9EgClPrYogbWkQxNcSeqIxxJ0LqE3xLE7L7VdbrAISfroq/Sx2zMPACtPJybWt1EdX5gblfgpI7v6Vu7gmP8IJhEz8+nebQtjVu4dREwhTbEjIODc8Ka2LxrLYgEhKyj0AyZ0cqVg8hqqxruR/Et+dbd7ZkjUeBSLWtZAvPiR6I374Q4CR8SjkRLUP6HVjm2RjDJQCHdCdFrf5OcJ/G+s62uyiyOsFhd98qJULnFUdEiKC6WdAbyco26fsHlQF+M08KO6DFWp9Hcxao/9cNP+ZYVD6abWbkmHiAjim/Two6Wb6xOk+IuWv2pHS48ajgEaQgUwSYtilyDqBZNkdGmhB0j39j7ruUu1qgBtHzK7KU6eTSKlhNyGAZb5IQd18FpX3MXZ9QoWkRFZIUuEXJQaDhaNNqMsJNDq7mLU38aRQt8Y1CMnEiW9y1AuGLvlA/NXRq8DGQy2Lr3/iMoBqNFjxw8F7c5YSfrW8NdGrwMZKvDWMSKiFKZEXJIdtGQb7xnEnfTwg5TE6vTpLiLNqo7EnKVV7O3lFwzJw4i+gTJMSLOGGHHUBOj0xjuYo3qBpufL5IrUrHY0DSAveen3MXp60TQDrpDffs8Bd8OQJglO+34a+Pee3Wqhr8BRm2cG54Z5jgAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
⇒ tanθ =![](data:image/png;base64,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)
⇒ sinθ =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQoU5WOyRKAIAxDAyru+4KK8v+fKS705DCQ05smTQs4pefE+jKviIEljPUU71/RwowK91kP924x8kgGRGTGeP3mdTdg5SMtq5T46Bs7nlhEDGyifYyzHKE+hhTU+fPJBSA0A0nazM4sAAAAAElFTkSuQmCC)
L.H.S
tan2
– sin2![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAXCAYAAADtNKTnAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjLY2hsbGSgFDOMMEPq8yJV3Y3lZjEwMPxnYJB9Y+yWl0KSIfV5HoYyMjJHPaLzvED8vGjTCJBBHnn1hkQZAjJAlkH2JrKG+vpYSWMGhruybjnFBA2BKWYwjlqIaTDDG6IMyXGTLcbm7IZoYx9Q2BA0BJcr8BmOwxWgmMCFje/G1tdL4jSkoyONx0OGYQ82hTCvYHMhA7aAY5Dx2JPW0cGDLBdlzLAQm1cwDMEVcPgMx2mIcXSDD7GuIMoQmCvQDSYcJtDAy4v28ALxTaPzIkjKgKD8ATYIFBOyRqci8+pVRwulATIEAEbNNLr35TINAAAAAElFTkSuQmCC)
by the above values of tanθ and sinθ
tan2
– sin2![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAXCAYAAADtNKTnAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjLY2hsbGSgFDOMMEPq8yJV3Y3lZjEwMPxnYJB9Y+yWl0KSIfV5HoYyMjJHPaLzvED8vGjTCJBBHnn1hkQZAjJAlkH2JrKG+vpYSWMGhruybjnFBA2BKWYwjlqIaTDDG6IMyXGTLcbm7IZoYx9Q2BA0BJcr8BmOwxWgmMCFje/G1tdL4jSkoyONx0OGYQ82hTCvYHMhA7aAY5Dx2JPW0cGDLBdlzLAQm1cwDMEVcPgMx2mIcXSDD7GuIMoQmCvQDSYcJtDAy4v28ALxTaPzIkjKgKD8ATYIFBOyRqci8+pVRwulATIEAEbNNLr35TINAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
R.H.S
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG8AAAAXCAMAAAAyauH+AAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZrbbZrb/kDoAkDo6kGYAkGY6kLyQkLbbkNv/tmYAtmY6tmZmttvbtv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bfWRMuAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABbElEQVRIS+1UXVODQAy8UFu0iiIqrR8FLVKFO+7//zyTgECB4wRnfLF5uIHhspvdJAjxb+MjAPB+o34aQHH7IlJnN59wBoA8r/j0E2rNpnNPBEhuSgq9WWXKX9n4cgg7VwYA6hL6aOl3+3LylY/R6PENAZhhknpaoiV6Kd1u9Ta9QwAxYCz2Or3gacwBwneX5zKnV6ZAO/Es/MrdHo3aAjheRrl2ANZHDdIRPUl3vRMJJhY+VdLiK1kHQm+WmX5EC9gBG0DjZ3Wdc46klNQABj4WLi9rvnGAIb6OdQWPZq0vKtkxqgmKYf1GupuCRwBKPn24c9k9zulct/VPYPPP9iN8bYC6f+byBM9n4Y/M58FFC4z62gDNbpmvJ7x/C5QwFNQ6RvwRAM3G6zMe6p6s5J3tzAtKo/+LaR2k+yBUgPo41woQgxMKXEPv04dQUhdpHo/BVQDOtWm91RaTrkge5s4BsP03Tt9PDvyRA1/q8S1W3PwvWgAAAABJRU5ErkJggg==)
by the above values of tanθ and sinθ
tan2
× sin2![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAXCAYAAADtNKTnAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjLY2hsbGSgFDOMMEPq8yJV3Y3lZjEwMPxnYJB9Y+yWl0KSIfV5HoYyMjJHPaLzvED8vGjTCJBBHnn1hkQZAjJAlkH2JrKG+vpYSWMGhruybjnFBA2BKWYwjlqIaTDDG6IMyXGTLcbm7IZoYx9Q2BA0BJcr8BmOwxWgmMCFje/G1tdL4jSkoyONx0OGYQ82hTCvYHMhA7aAY5Dx2JPW0cGDLBdlzLAQm1cwDMEVcPgMx2mIcXSDD7GuIMoQmCvQDSYcJtDAy4v28ALxTaPzIkjKgKD8ATYIFBOyRqci8+pVRwulATIEAEbNNLr35TINAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAAqCAMAAADyHTlpAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OmaQOpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bvF3shwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAyklEQVRIS+1U2RKCMAxsvKgKHmi02GL//y9pCy3oQA8dxxf2gekMm2SzSUvIP1AdrnFl6wKW9yiq2N1YJFXlm6kpDizipiUvFABWx6h5zaRXB1BZ1yLS7TcDXfjgkOTxtwKSiv2EzNrWNw8iqwJg7dlaPCsFgqovwonUeejlMLcFM/1ygA6dxjPf2588QGX9pNEvQJa6tIER7UFfVJZOyTgflVMGg/TjTEFtKsy6mCkJrhOms3OPBOeU2KpJSfRQuXWqW9JAYx9vVwNapgkZb7/bugAAAABJRU5ErkJggg==)
Question 12.In ΔABC, m∠B = 90, AC + BC = 25 and AB = 5, determine the value of sinA, cosA and tanA.
![](data:image/jpeg;base64,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)
Answer:AC = 25 – BC
AC2 = AB2 + BC2 (by Pythagoras theorem)
(25–BC)2 = (5)2 + BC2
⇒ BC = 12
⇒ AC = 13 (by AC = 25 – BC)
By Pythagoras theorem
(25–BC)2 = 52 + BC2
⇒ BC = 12
⇒ AC = 13 and AB = 5
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 13.In ΔABC, m∠C = 90 and m∠A = m∠B,
(1) Is cosA = cosB?
(2) Is tanA = tanB?
(3) Will the other trigonometric ratios of ∠A and ∠B be equal?
Answer:(1) yes, because the side AB and BC will be equal by property of triangle and therefore all the trigonometric ratios of these two angles will always be equal.
(2) yes, because the side AB and BC will be equal by property of triangle and therefore all the trigonometric ratios of these two angles will always be equal.
(3) yes, because the side AB and BC will be equal by property of triangle and therefore all the trigonometric ratios of these two angles will always be equal.
Question 14.If 3cotA = 4, examine whether
cos2A – sin2A.
Answer:![](data:image/jpeg;base64,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)
Let AB = 4, and BC = 3
⇒ AC = 5
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
R.H.S =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHcAAAAXCAMAAAAGEyDiAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZpCQZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tmZmttvbttv/tv/btv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///b1MdH3QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABm0lEQVRIS+1V23aCMBDcWGtarWKtYIVeU9sC4f+/r7tLgkkE4eBLH8xD4lFmZnd2ggDX5TnwsxFieYkn4wj0eg+HSTpe+AKC4u4CXap4JIFa2Xar5Pa3t/VcxMEzRwIYxFDDD8fx5mKA5Se6DgEMYmBZ5aQqe5JN77192wdcAggYCkztBO2hc/GOiHKHXyzR1Bxlc2Ocfvw4Y7SB5AJ9pu1L8kVwCSBgKOQWyigGPpObTx5D9YzD1JHAZXTVDBT+2L4aSCGpATlPQSHQI4CAITNxyWaUPsTpCP0sFn6IqoQqC0Njq2ggRpfFg6kEDAzBVZ+8v4j5W9gX1VElVBqtjIzgZbJmIUddy9sQ+QxGz+rW1Id7MQ0sVazSabSFnNENGFr6ReVvaVur69Vren2c3k7HFoZ064YM1rz6UhOOR2unbohzDoGO/Go8DxnSrXvCoMQWqtcUm1lhnhmLwd54CqY4HXUY3UDYEd6CXLUw4F2b7mmqEv99sK1yhx8e3DjTfTD3ql3YQgpExrwRxAl0L8PgN9H1wasD/8yBP6EUKfsr+Pq8AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAAqCAMAAADyHTlpAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bRdydOgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA20lEQVRIS+WU2RLCIAxFg0txwQWNokLL//9lE+o49aGQ1tEX89DpDIeS3HIvwM/LqVTLR/lkPBITND9F5eZXEQfQmK2QBDc7C9FoKyEJXomHQolS6dygxUPhN5TyYqWkMv0Ph50rqKaJ+Nreexml3qcNjDpsOnwbNmO875Ra8HrQBSFRHaA27IFiuiB72rFdi2iaKjlbhiYTitAOCnqtuwGHK9qUAfFEA9rsVXpLq3x0YNUL68ZkUs5xWHlqAbmN3FfDigaJjCG1WdtMID2vJqH1nn7s5jL9+uR3tj6ICYjG/mldAAAAAElFTkSuQmCC)
Question 15.If pcot θ = q, examine whether ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANcAAAA6CAYAAAAjp8zpAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAZWSURBVHja7VwxctQwFPUBuAAzK7ocIGtS0jHGk0nNkHhSwVAxnmGoGZMuDRegS8kJKKClyQ0ouEHuACt7ZWsVyZbkL1v2vuLNJJtd++v9/6X//zovubm5SQAAoAdIAAAkFwAguQAAQHIBseL29v2TfJP8SpLkX5Kkf6+r6unS7IEjgShRZum7vKxO+c9XaXKXbPJf729vnyzJHjgSiB5VmZ+yZHs/9+nlag+cB8SfXNX1022y/RlNclnaM5uBVyn7ymvXy7I6od1VLk9epc++NbUxe0iz8h0CdNnc8JMiJltt7VlVcvFFbzab33lRntd1cnH2hgeRqJWPvbRaKjdFnr+ds9/ytWdldTn7IwcLP77TJPnLsg+f0LMsk5sPGfuUFl8ulmjPampyHihJenX3OKiSh2NOriVzwwNZ2MfXkWXl6yXZs4oA4ovWlThfivSC9xfHnFxL5aYed9e9ocC8JayPPU4X5Y7gO96WJff1a4ZZfz1N2b/nWfrqW1lenqZpUamNtRhn8i/oivzsI//9sipPmn5sd322vR9agGln7gss8uHAzs7r8vqFyQ7x/pa3PS8qd/w9zxn70TpQs/4hbmPghjqeJh/2WPiTrCxsdjn2kGXp57OifCNeM5UVnEAxTeGJUxO6N7J1uPRNd0c4ezjLi4+cYFG2DC2uCRJ5R1FB/+2+bBu3tVljsyHo6vGGq13Psx8mtJ+Xgkkt0+RklK/Zx20M3ISIp2l6Unt/kiaXCH554cZafv+6bFT9mrK7NtfsnKv+3j5u0rObdY+kPA4SUfaM2Xl6HaHYpdqv2ihz1zpvdw1xcuvW2t5rf11bbufixvX0somnyRLL0p+kydUu2uB4daeRn8HiZYtp1E6RXCZiuuvpy57H9bMe5rUdkt4XGMLGvh3QdEp392vWYcvtGG4mG7BYxpNfPxTOn7TJZQgOsfvpgkbtL3S9A0VymZryvsCi2OXU+/Vx0fc3m5NEBI74vA23c3DjyqFLPIW2xcWfpMnV1O2HmW0T+I+aROUalMmlkhBqZ9YNAfrKr75Taewp08ftHNy4cegXT1NMUof8SZZc7Y2UwOibNPHPZGn2WSZJ5+hQyWVTho3rFQ5JL9K0Yiz5Y5wSSgMclheVvBb+pDW3Uy73ZLtVDmy5nYMbG/jEU/jez82fZMmlHpvNyJxVfUS0DXx69VUEQfOojSZx9s5Xe4vDoDTvIOqpUBb5Of9dTKDC7HSdzdwReVmcDz3p0JZ9KqSNo3tP90iYGnQ23M7FjU8ZZhNP4U8ud3+SJJdulDvUSNcBkBdvy/Lli7Y3kPqCbgJmajw50c0CbUbGPLja6+3uQ/0gsGlYw1j+nd9LjJWHgqMObr4j9vB48J1PfY/nP+T3DHE7JzcuwewST5OcpB7+HJVcpiMc0JQWM/8j3xKwlHii9OfgyDSmhyZjAx4Mducq5nii9qfTVAcYnjYBy40nan+ahwRSXYzTa7h3AEf9Q4yYuQrlTwQAAAQCSAD6m3uPx4oAJBcAILkAYPHJZVMCAOtEkOA6cg6xwwDouVAWAseE2LTifWyCI4EoEZtWvI9NcCQQPWLTire1Cc4D4k+uyLTibW2aqVEOoxPf7CjQil8bN7FpxdvatKrkglb8OrmJTSve1qaV1eXQil8bN7FpxbvYtJqafIweeow1fSzczB3EPlrxIf3pYtMqAmisHvqak+sYteJD+dPVJqcLUmjFqzrx/LW5teJ9neGiLw6teP948jmtQ/vTBlZvotKK1+nEHxI+j1a8jzNc9MWhFT8unqZIrhB68U67DZVWvE4vbk6teFdnuOiLQyt+XDxNkVyh9OLtG2JCrXiq5AqtFa9ztKu+OLTix8WTXy8Uzp+0yRVAK54quaj00F12Old9cWjFj4+nkCdXSL14y2kTrVY8dXKN1UN3cYarvji04uniKURyhdSL7/1jKK34kMnlo4fu4gxXfXFoxY+Lp9DJFVIv3unIpNCK1wXS3Frx7judm744tOL942makyuMXrzVjam04nUadnw37/4BbR6teBdn+OqLQyt+Oq34Kfw5KrmgFe84rYJevFdJuGZ/Do5MoSRrN1qGjoQdT7HHE6U/naY6gN20CVhuPFH60zwkgA66c/8Anm4WG08h/IkAAIBAAAkAgOQCACQXAAA7/AfhBzJ/ag/eyAAAAABJRU5ErkJggg==)
Answer:L.H.S
dividing by sinθ
⇒ ![](data:image/png;base64,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)
substitute cotθ =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAhCAMAAADambo9AAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6Ojo6OmaQOma2OpDbZgAAZjoAZpDbZrb/kDoAkNv/tmYAtmY6tpA6ttv/25A627Zm2////7Zm/9uQ/9u2//+2///bcMixrQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAeklEQVQoU62QSw6AIAxEWxT84l8RBO9/TAkicUFk4yQsJryhHQC+pTvEit3MOTJpz22OcgIQMWN4I/WDgSqwXj3mgk/GmZnKMF8htolt0CuB/XutkK5Iev+oyCbYie3uuuQSDB9iJttABMyuTTzlMkHLy9j+73+KlLgAjJQFMgYWZYAAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
= R.H.S
Question 16.State whether the following are true or false. Justify your answer:
(1) sin θ =
, for some angle having measure θ.
(2) cos θ =
, for some angle having measure θ.
(3) cosecA =
for some measure of angle A.
(4) The value of tanA is always less than 1.
(5) secB =
for some ∠B.
(6) cos θ = 100 for some angle having measure θ.
Answer:(1) No, it is false, because we know that hypotenuse is smallest side and it is the denominator which is smaller than the numerator.
IT IS FALSE.
(2) Yes, it is true, since the denominator is greater than the numerator which implies that it is possible for the hypotenuse to be the greatest side, and ∴ the triangle could be formed.
(3) Yes, it is true because the hypotenuse is the numerator in case of cosec and it is greater than denominator which means the triangle can be formed.
(4) The statement is false.
The value of tan could be greater than 1.
consider a triangle whose tan of an angle is smaller than 1, then the other angle will definitely have a tan greater than 1. because their ratios are just inversed.
(5) The statement is false, in sec ratio numerator is the hypotenuse which is the smaller than the denominator which is not possible.
(6) The Statement is False. Because here numerator is 100 and the denominator is 1, and denominator is the hypotenuse and the numerator is any of the perpendicular side, and hypotenuse is always the longest side. But here it is not so and hence it is not possible.
In ΔABC, m∠A = 90. If AB = 5, AC = 12 and BC = 13, find sinC, cosC, tanB, cosB, sinB.
Answer:
we know that
Question 2.
In ΔABC, m∠B = 90. If BC = 3 and AC = 5, find all the six trigonometric ratios of ∠A.
Answer:
Pythagoras theorem
32 + AB2 = 52
⇒ AB = 4
we know that
Question 3.
If cosA = find sinA and tanA.
Answer:
Let AB = 4 and AC = 5
BC = 3 (by Pythagoras theorem)
we know that
⇒
⇒
Question 4.
If cosec θ = find tan θ and cos θ.
Answer:
Let ∠A be θ
BC = 5k, AC = 13k
⇒ AB = 12k (by Pythagoras theorem)
we know that
⇒
⇒
Question 5.
If cosB = , find the other five trigonometric ratios.
Answer:
Let BC = k and AB = 3k
⇒ AC = (Pythagoras theorem)
we know that
Question 6.
In ΔABC, m∠A = 90 and if AB : BC = 1 : 2 find sinB, cosC, tanC.
Answer:
AB = k and BC = 2k
AC = (by Pythagoras theorem)
Question 7.
If tan θ = , find the value of
Answer:
Let’s divide both numerator and denominator by cosθ, we get
put tanθ = in this equation.
Question 8.
If sec θ = find the value of
Answer:
Let ∠B = θ
⇒ AB = 5k, BC = 13k
⇒ AC = 12k (by Pythagoras theorem)
we know that
⇒
⇒
putting the above values in the given equations.
=
Question 9.
If sinB = prove that 3cosB – 4cos3B = 0.
Answer:
Let AC = 1 and AB = 2.
⇒
∴ BC = √3 (by Pythagoras theorem) we know that
⇒
⇒
=
= 0
which is equal to the R.H.S
Question 10.
If tanA = √3, verify that
(1) sin2A + cos2A = 1
(2) sec2A – tan2A = 1
(3) 1 + cot2A = cosec2A
Answer:
Let BC = √3 and AC = 1
⇒ AB = 2 (by Pythagoras theorem)
we know that
⇒
⇒
⇒
⇒
⇒
(1) sin2A + cos2A = 1
⇒
(2) sec2A – tan2A = 1
⇒
(3) 1 + cot2A = cosec2A
⇒ L.H.S
⇒
R.H.S
⇒
L.H.S = R.H.S
Question 11.
If cos θ = , verify that tan2θ – sin2θ = tan2θ⋅ sin2θ
Answer:
Let ∠B = θ
and BC = 2√2 and AB = 3
⇒ AC = 1 (by Pythagoras theorem)
we know that
⇒ tanθ =
⇒ sinθ =
L.H.S
tan2 – sin2
by the above values of tanθ and sinθ
tan2 – sin2
R.H.S
by the above values of tanθ and sinθ
tan2× sin2
Question 12.
In ΔABC, m∠B = 90, AC + BC = 25 and AB = 5, determine the value of sinA, cosA and tanA.
Answer:
AC = 25 – BC
AC2 = AB2 + BC2 (by Pythagoras theorem)
(25–BC)2 = (5)2 + BC2
⇒ BC = 12
⇒ AC = 13 (by AC = 25 – BC)
By Pythagoras theorem
(25–BC)2 = 52 + BC2
⇒ BC = 12
⇒ AC = 13 and AB = 5
we know that
⇒
⇒
⇒
Question 13.
In ΔABC, m∠C = 90 and m∠A = m∠B,
(1) Is cosA = cosB?
(2) Is tanA = tanB?
(3) Will the other trigonometric ratios of ∠A and ∠B be equal?
Answer:
(1) yes, because the side AB and BC will be equal by property of triangle and therefore all the trigonometric ratios of these two angles will always be equal.
(2) yes, because the side AB and BC will be equal by property of triangle and therefore all the trigonometric ratios of these two angles will always be equal.
(3) yes, because the side AB and BC will be equal by property of triangle and therefore all the trigonometric ratios of these two angles will always be equal.
Question 14.
If 3cotA = 4, examine whether cos2A – sin2A.
Answer:
Let AB = 4, and BC = 3
⇒ AC = 5
we know that
⇒
⇒
⇒
⇒
=
R.H.S =
Question 15.
If pcot θ = q, examine whether
Answer:
L.H.S
dividing by sinθ
⇒
substitute cotθ =
⇒
⇒
= R.H.S
Question 16.
State whether the following are true or false. Justify your answer:
(1) sin θ = , for some angle having measure θ.
(2) cos θ = , for some angle having measure θ.
(3) cosecA = for some measure of angle A.
(4) The value of tanA is always less than 1.
(5) secB = for some ∠B.
(6) cos θ = 100 for some angle having measure θ.
Answer:
(1) No, it is false, because we know that hypotenuse is smallest side and it is the denominator which is smaller than the numerator.
IT IS FALSE.
(2) Yes, it is true, since the denominator is greater than the numerator which implies that it is possible for the hypotenuse to be the greatest side, and ∴ the triangle could be formed.
(3) Yes, it is true because the hypotenuse is the numerator in case of cosec and it is greater than denominator which means the triangle can be formed.
(4) The statement is false.
The value of tan could be greater than 1.
consider a triangle whose tan of an angle is smaller than 1, then the other angle will definitely have a tan greater than 1. because their ratios are just inversed.
(5) The statement is false, in sec ratio numerator is the hypotenuse which is the smaller than the denominator which is not possible.
(6) The Statement is False. Because here numerator is 100 and the denominator is 1, and denominator is the hypotenuse and the numerator is any of the perpendicular side, and hypotenuse is always the longest side. But here it is not so and hence it is not possible.
Exercise 9.2
Question 1.Verify:
cos60 = 1 – 2sin230 = 2cos230 – 1 = cos230 – sin230
Answer:cos60 =
, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQoU2NgwAskBVhg8kJs3HA2A4MgaWxJfmZRqEGCjEDAgd9aImRBpgABESpJUCLCzcjIDlEvwcXLIMzEB9cszopgCyKcLwxVDlQniGCKAZliPBBzOEHOhLCxAACkwwJsDMBvegAAAABJRU5ErkJggg==)
2cos230 – 1 =![](data:image/png;base64,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)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQoU2NgwAskBVhg8kJs3HA2A4MgaWxJfmZRqEGCjEDAgd9aImRBpgABESpJUCLCzcjIDlEvwcXLIMzEB9cszopgCyKcLwxVDlQniGCKAZliPBBzOEHOhLCxAACkwwJsDMBvegAAAABJRU5ErkJggg==)
cos230 – sin230
=![](data:image/png;base64,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)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQoU2NgwAskBVhg8kJs3HA2A4MgaWxJfmZRqEGCjEDAgd9aImRBpgABESpJUCLCzcjIDlEvwcXLIMzEB9cszopgCyKcLwxVDlQniGCKAZliPBBzOEHOhLCxAACkwwJsDMBvegAAAABJRU5ErkJggg==)
Question 2.Verify:
sin60 = 2sin30 cos30
Answer:cos60 =
, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
2sin30 cos30 =![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Question 3.Verify:
sin60 = ![](data:image/png;base64,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)
Answer:
and ![](data:image/png;base64,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)
L.H.S.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAAvCAMAAAArSD+zAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAA///b//+22///tv///9vb/9u229v//9uQ29u2ttv/ttvbkNv//7Zm27aQ27ZmkLbbZrb/kLaQZrbb25Bm25A6OpDbOpC2tmY6tmYAkGY6kGYAOma2OmaQkDpmkDo6ZjqQAGa2kDoAOjqQZjoAOjpmOjoAADqQADpmZgBmZgAAOgA6OgAAAABmAAA6AAAA3+7SwAAAAAF0Uk5TAEDm2GYAAAHXSURBVHja7VddV8IwDE3XWSYysOIqQxA3B0Ms5P//O0+z8o0chUZfzEPPxnZy15ubmwLwH38RGW6DI7+YdHg3EL1KXoC0YC5BrgFEb4Y456FKvMQA+eoeVLliQUjeJEBekJo0i0jNhqtlHDb1TcutA8+LGo7bQdOrITpeonf67GiBOA4r19sWUbIVadeGpiixBiDbFjZFE1ig5TLe9QkCDNvDqEmkIErT3IbusbLumeZi1QFV1sFNKcOZ9pKyiKPwphctQgvnaAvFuQIdMjqwRKIYIr61Qyjg4Ac7avnKSVWG33pq+/7Cme4VilMTxNVIpoga3PJocUTlKnY9cXt7gXSleK4lJFa73pt2GhvffLIgNUeLSzUdLUwzIzyA9h2eLx8qUnIDIPaaJt+cSr4xswY4vWuMYw3gMEWJTxK61pwEOBl4MgCgN8N5fAxACfNarim65qjQtcUXANkyXhd51xh/QlEzomt5AODnqXuwX/MLhkQfVFX4FLRQkRNbSOihbgQkLjdGNbE0PBOLqGlx09QhVIjze3qlCj5er5oJ3YrtTOcF0AfF6dqcZ7pz3h96G8yDjdqJU0ilYc7P/BcnL3i7MnOukvKRlHx0AERuGDuZgrfMvxmfcBY4Xri2g3wAAAAASUVORK5CYII=)
R.H.S
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 4.Verify:
cos60 = ![](data:image/png;base64,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)
Answer:cos60 =
, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFwAAAAhCAMAAABjjTx1AAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjpmOjqQOmZmOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmY6Zma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJCQkLyQkLbbkNv/tmYAtmY6tmZmtpBmttv/tv//25A625Bm27Zm27aQ27a22/+22////7Zm/9uQ/9u2//+2///bm8RAVAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABr0lEQVRIS+1V7XaCMAxtcB8y51R0XzC2CeoEnVLo+z/bkhRwO0PFCv+8P2KPprfNzU0V4oJSATXrtqZG/DBpj1yI6EJe1bo2ZVHh7faoXdSqB3AdUJ7yAfpfR3dwQgSIUZ4r73j/f4TWVKSehb8q72abep1lPfbfWQltryQnR0XgCsEpCa1OxIyq6CxZgyFRgBvbtNII6b4cMsfAvnwt5XW3KqSVtO8DfV9E6g9QaVKFyCmeiJ0s0kbKPFBHMgdggISaXMeyIiqYsU9Vzqwiz5y83WsbFakir6ygPDBfFORq/Wij3PnNS3LswaiU5RzNC0XwhB25tLGGvKGFebnRJ8jC2uw0J3LlERkbUPvRwIpIORLzTwzpEzWUOeg7RfOTlkb528+DpuEZJWDWDCwXAww3DriSlCeb4GE+rofskHSifVMPc4NRrseMDXuum2mQl7wIEffAwo/mEQdCvU3F6uB0GR6r3nVz9r62prwLlEJqyVP/1ZBlz7bvBT4/Cb95IVw1TI6z4YrCiGuamkYR3WxIlWwc6HlvFNn4gx2I/zctWDFqw4BF/bJf+5U4U7IfRyAqoozPhhYAAAAASUVORK5CYII=)
cos60 =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQoU2NgwAskBVhg8kJs3HA2A4MgaWxJfmZRqEGCjEDAgd9aImRBpgABESpJUCLCzcjIDlEvwcXLIMzEB9cszopgCyKcLwxVDlQniGCKAZliPBBzOEHOhLCxAACkwwJsDMBvegAAAABJRU5ErkJggg==)
R.H.S
![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Question 5.Verify:
cos90 = 4cos330 – 3cos30
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
R.H.S
![](data:image/png;base64,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)
= 0
Question 6.Evaluate:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMoAAAA0CAYAAADL/afBAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAgqSURBVHja7Vy9btxGEN70dm8Z3uvsXrenMq4MmvFPKSMUcZUVFcKBgOMAKoSAUick8gPYld05hdsEiF2msZ/ABqQnkJ7hEu7yllwu94/kkkcpU0whisedmZ1v95sl8aHj42MEBgZmNkgCGBgABQwMgAIGBkABAwOgyE4g9B031/ts917ZCYEc9F5TbXLYi3NNfpMm4SbB6HP2uyVC+JIEi59U9x0c7N3cIfhVfh9aYrLzau/g4OZ1mtzD/QdTjNAliY+eqP5/FJMnWez/iqa79/8MkEUUPJ5h/BerFRx+3Ds5udG1jrw5mKbzDYLQGSLx2yYg2QrjFzSQk5O9G3GIU93k7xD0jgdN7w1n6AMiO+98rdQ6o2MFJPhVTHYfxmK6gz7p4hf+vyyNnM3TdAMAUoIgzxG+wCQ+TZKjW77qaG1BUSfnwfyZshgkx/OVFF+GSbopgsy0+sognqLpxzZFtQjwSxwsXvadDxpjFs+FDigsB1legG7pQDK/nS3U53TxiJL0rn5HbldH3jmf6V4bNeNAkQuTrQJ3wk/iqq4DlS+gsC06vHNEE/tg/3BqopldY6f+zfDs/XxOdlVAEXebCXn4Znv/8N4QgHHh8k3vadunmsYp81MFgXI3aVlHztw5I89fOKfb39+ehlF6v0K5cjrAJjkm6C3/mxd92YtgLXoZR8+er6B056pAWOCInNsA0BQoaRTe56t7wWVXcVAAxeHWi9X/2XXRjzaxU85Mr/MeRL5ndZ1TLtabyHnqg8ZQv3icCE+/yPlL9/Y2RL5P76Egdq0dVT5VxW7zheeN5lsHpq515M6dswEoEumkTxH+IgdDKYo4WL6l4W/0PhoI7UWSNLqrcpYmYhFtRfT+IFo8lvsYujWq6E8+pnkVabuj8MSK4xYcGJPPfHtPArKLMfomxtQkdka5VkWvAwq3JIk2w9nktQzEvg4VJrPwNY+TFpM4XhqRp2xesl6g4Ps0N8KKbaudw/3texREfBzV6m7zpdxNyHkYh891B0Nd66hJwfzCr8WEpGLRqdDKBs+SliQPvg/j5JHOWXlHkgugTYDyNk35KwPKwcFt16NBmc8WEyKdoqiKu0nsjHKtcmkDipSTC90qKMeoM8OzL8VDGXptgiZfywLPi1c+uJFzZqqdoqcQnlECJb/m4ktRe9kuEyVJMVernb2yq/cKlPJB2QRG6VPTsSWfYB4w5dNBkDxzcegoSW7FBJ/mYJF3JnWAui1ToD9LibJUrxlO6GQ+q/NdByiX2Ok1ERSuQLHdq4m/ZtqcSjxe3Q/U866LUVU78tyxY92clRQneTZfTPVRAGg1Rps6atXM50WsbmzlgYqVQLUNWwLPn1VflQbtUSTapaJhusJyjZ3dl1EucYWndIYXla1Z5z75fI9iKiaXe/QLV7V2xDmlfU4Sb/04maCvIqV18cXNn7yWeu9RaiuJVOha2iWtdq6BaynPgKdesg+qY0W5iWwa++q+pX7nwxcmKpDHVO8Vux5R23Y03T0lHavPhzyHwovTJZpMvub9R7LZ1BfbQir2zV3ryJg42qjWTxeqBaOjXfLAqm1ZxZXZ85R9QPW3trfYXYCipl3VFSdNorshxn8oQe0Qe5MiVO0u+Y4Un/p+l6Man8W66rVUcyG+6OPXTbVjijMOw+cimEy+VKlmNj9ZD1qbR2EuutSRdQfBYZzSV/z5ER3boipvg3lRsVMdEqeqFzgimmkiRM4ovhdI4vARxvgfuaDY7zH6yE5PMl/oaYnY9PkEikizaGJpHHyX2IoWEfeTBMkuj30+D55RyuAae5NCpb9/SCZvSBj9TGMvxifh776/FigOCegiIYw1m4W/FZSx+AIjn4uyLkqQ2GqneEY2Dp97CoAfMH4vNd9GX2oHQsK9q8Wpsit3qSNjwdBmtPItlsAhq6cL+IJ/ipJPdhVMRSDC7wunC8qh/+ygHIs1+0txPPeeg/ztBJSEHeOeYTz7k69cFV+zGPh1Hjt/L+AauwUoS3l1W33as+RUpUnsTY32C8I7jQu6IKgAJRzD1ubNpXbkZ4jHv018qYCgPDpf0vlT5bttHcHnD2BgDgZJAAMDoICBAVDAwAAoYGAAFDAwAAoY2DUHiutXp2Bg191sQFmCgYGhJVAvMDDoUcDAAChgYAAUMDAAChgYAAUMDIAyHkdByNs5H9c9B+uogSuRGBDyrhZBHyLUV8XWJWR+JUAydiHvoaxPEeqx5MAkir5OIfNRg+SqCHkPA5J+RajHkgOTKPo6hczX0jt04ZVjEvJ29b2rkPUQItRdcuBDzNsmir5uIfPOfNEkwMwT0FWEucZRRyDk7eq7DyHrIUSo2+bAh5i3SRS9jGG9QuadaJFNvNuHCLM4IWMT8rb57kPIeigR6jY58CXmLQJd5f8YhMzbN9kW8W5fIszCWKMS8rb57kPIurJT9CBC3SUHPsW8ddcqlMuzkPmg1Msk3u1LhLmS4BEJeZt89yVkbYrNhwh1lxz4FvPWPa8vIfPBm3mVeLdPEWZtka5RyNvmuw8ha/dn+RGhbpID32LeOto1JiFzL0e4Muf0KcLscgw6tJC3zXcfQtYuTbpPEeomOfAt5q2jXWMSMm/1I5t4t08R5jEKedt89yFkXadI/YpQ+wBKGzHvphR8XULmrXcQk3i3LxHm0Qp5W3z3IWStoBC9ilC3oF6dxbxl2sVF0ZsCZQgh81anXTYBZp7MriLMYxXydvHdh5B1LRc9ilA3zYEPMW++C8mi6A5AGVzIfHSfaoCBjdEgCWBgABQwMAAKGNhg9h+mKU0Q8KJDlwAAAABJRU5ErkJggg==)
Answer:cos60 =
, ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Rationalise the denominator
![](data:image/png;base64,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)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEYAAAAkCAMAAAADzLznAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OgBmOjoAOjqQOmZmOma2OpDbZgAAZgA6ZgBmZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkGaQkJCQkLbbkLb/kNv/tmYAtmY6tmZmtpBmttv/tv//25A625Bm27Zm27aQ27a229uQ29u22/+22////7Zm/9uQ/9u2//+2///bTCGwIgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABgUlEQVRIS+1Vf1PCMAzNUGSKDhDUsanohkg3f7HRfv+PZtomW+dfrud5nGfvuNLw8pq8pAHgENf+7tEzrPJq13jKxXNPFhHQgvq84cmXPVngadt4iAl9rc9a4/f45HWLk5fkLUbWWIc6qmIcDG4ctvd5EER41nkMMvtDhQDGqYRyyW1YKh2jQaUPUDIcrXJGZ+FkXmQOjt2JTsTMW5/SrZy1Pjs06t7KanGUDEVVRRze/jbuSmRU1ElFoDaYT22lIVyHRi62Kon1NXlwHIOc2orqTEotDYn3uhnuoDL5GdzXaGwnIAbXi9Gal2AWUMkEP8um3BbHFSKNDAIlzahkRFMhC16vUrKL4YfOqcWxO9O9hUfYAkXYKbjNDmnWWG+dhJytTD8wril4z/YTTjtoWm4/6PcYnFdkFObHAP2fZlsAl/RnBkW32TxPPDr8d8+L/90OTgG1tmOad78Ai4u5oeHdj8UZSTxLPInY/S/RqPzEzHje/aQx/xY43Hj3Y/ltr0+d/yZ154RkJwAAAABJRU5ErkJggg==)
Question 7.Evaluate:
![](data:image/png;base64,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)
Answer:cos60 =
, ![](data:image/png;base64,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)
denominator =![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 8.Evaluate:
2sin230 cot30 – 3cos260 sec230
Answer:=![](data:image/png;base64,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)
⇒![](data:image/png;base64,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)
Question 9.Evaluate:
3cos230 + sec230 + 2cos0 + 3sin90 – tan260
Answer:put all the respective values
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 10.In ΔABC, m∠B = 90, find the measure of the parts of the triangle other than the ones which are given below:
(1) m∠C = 45, AB = 5
(2) m∠A = 30, AC = 10
(3) AC = 6√2, BC = 3√6
(4) AB = 4, BC = 4
Answer:![](data:image/jpeg;base64,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)
(1) If ∠C = 45 and ∠B = 90
⇒ ∠A = 45 (angle sum property of a triangle)
∠A = ∠C
⇒ AB = BC (sides opposite to equal angles)
and since, AB = 5
⇒ BC = 5
and by Pythagoras theorem
AB2 + BC2 = AC2
⇒ 52 + 52 = AC
⇒ AC = 5√2
(2) if ∠A = 30 and ∠B = 90
⇒ ∠C = 60 (Angle sum property of triangle)
![](data:image/png;base64,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)
put the known values
![](data:image/png;base64,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)
![](data:image/png;base64,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)
also, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(3)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ∠C = 30
⇒ ∠A = 60 (by angle sum property)
also, by Pythagoras theorem
AB2 = AC2 – BC2
⇒ AB = 3√2
(4) Since ∠B = 90
⇒ AB2 + BC2 = AC2
substituting AC = 4 and BC = 4
we get
AC = 4√2
Also since, AB = BC
⇒ ∠A = ∠C (angles opposite to equal sides of the same triangle)
∵ ∠B = 0
⇒ ∠A = ∠C = 45 (Angle sum property)
Question 11.In a rectangle ABCD, AB = 20, m∠BAC = 60, calculate the length of side
and diagonals
and
.
Answer:![](data:image/jpeg;base64,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)
∵ ∠BAC = 60
we know that
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
since, diagonals of any rectangle are always equal
⇒ AC = BD
Now
by Pythagoras theorem
AB2 + BC2 = AC2
⇒ 202 + BC2 = 402
⇒ BC = 20√3
Question 12.If θ is measure of an acute angle and cosθ = sinθ, find the value of 2tan2θ + sin2θ + 1.
Answer:cos θ = sin θ
⇒ tan θ = 1
now ∵ θ is an acute angle and tan θ is 1.
⇒ θ = 45
![](data:image/png;base64,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)
now by substituting ![](data:image/png;base64,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)
we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 13.If α is measure of acute angle and 3sinα = 2cosα, prove that ![](data:image/png;base64,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)
Answer:3sinα = 2cosα
![](data:image/png;base64,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)
L.H.S
by substituting
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 1
Question 14.If A = 30 and B = 60, verify that
sin(A + B) = sinA cosB + cosA sinB,
Answer:L.H.S
= Sin(A + B)
= sin(30 + 60)
= sin90
= 1
R.H.S
substituting the required values in
sinAcosB + cosAsinB
=![](data:image/png;base64,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)
= 1
Question 15.If A = 30 and B = 60, verify that
cos(A + B) = cosA cosB – sinA sinB
Answer:L.H.S = cos(A + B)
⇒ cos(60 + 30)
⇒ cos90 = 0
R.H.S
cosA cosB – sinA sinB
⇒ cos60 cos30 – sin60 sin30
![](data:image/png;base64,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)
Question 16.If sin(A – B) = sinA cosB – cosA sinB and cos(A – B) = cosA cosB + sinA sinB, find the values of sin15 and cos15.
Answer:substituting A = 45 and B = 30 in
sin(A – B) = sinA cosB – cosA sinB
⇒ sin(45 – 30) = sin45 cos30 – cos45 sin30
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 17.State whether the following are true or false. Justify your answer:
(1) The value of sinθ increases as θ increases from 0 to 90.
(2) sinθ = cosθ for all value ofθ.
(3) cos(A + B) = cosA + cosB
(4) tanA is not defined for A = 90.
(5) The value of cot
increases as θ increases from 0 to 90.
Answer:(1) True
sin0 = 0
sin30 =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQoU2NgwAskBVhg8kJs3HA2A4MgaWxJfmZRqEGCjEDAgd9aImRBpgABESpJUCLCzcjIDlEvwcXLIMzEB9cszopgCyKcLwxVDlQniGCKAZliPBBzOEHOhLCxAACkwwJsDMBvegAAAABJRU5ErkJggg==)
sin60 =![](data:image/png;base64,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)
sin90 = 1
we can see its increasing with increase in the angle in the range 0–90.
(2) False
They are only equal at θ = 45, otherwise are not equal.
(3) False
Let A = B = 45
L.H.S
⇒ cos90 = 0
R.H.S
⇒ cos45 + cos45
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAF4AAAAuCAMAAACWLl6dAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOpDbZgAAZgBmZjoAZjo6ZjqQZpDbZrbbZrb/kDoAkDo6kDpmkGY6kNv/tmYAtmY6ttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bFnFZMgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABdUlEQVRYR+2W23LCIBRFQVtNr9beaC1Vam1Iw///Xw+R0DgT5OzM6INjXuIDa0sWGxIhTvlaP36gj8dHqrkcr7B4ALF3SwPGgwgaT4+KIMjYYBFBkLHn+HxNEZ/I2CHuR/CuNVzELQop5cVzXkgcMQAB0s9DvQGnXg8pAo63N0tgPnC8KLl9ReTUM6p2uIB8fPb0IhiF9fr/x+6v1p3um89+pEFjfH4RuLM/mpyDLG300F9M03HaVbZHTgrpMe7eEudxOj6J9MTXTz/5Zd4dgSDlFE0XCKJpI7j1XMpL9ksLQNw7fWFq+SKqGfdjEEHsLanXXpCRzAMYQcx9q77kxvOQ340P/ozGdV4OH6kW0hupH9qPe1vk3CDI96aZbeyYU9FSqqgYYgsKNGHKTnHqjyBOjVdxe+spZ+9CCFWl6Zgv5YTuZVaPQBCnJl/bRHtF/XE6Hw8hRl5v1Yd3Xz6e9h4fqdnnQCwThBhOXXZ7OgCBT+QjAn+zXyRlnKoP5wAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
(4) True
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJEAAAAqCAMAAACEC0mnAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpCcOpDbZgAAZgA6ZgBmZjoAZjo6ZmY6ZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLyQkLbbkNv/tmYAtmY6tmZmttvbttv/tv/btv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///b6T7I5QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAChklEQVRYR+1YYXPSQBC9RWujbdFUqhZqK6GtRhGS3P//b+7bS2AGWu4lnh0/cDOEMHPJPt7u7bt3zh3HCzOwlulORD97syJBLK/m5Mwe0/YRrWXExakn8upHj1CDpxafsw/Mw9XFY5kSUX0jMhqv1qJZw+VnJmPD0Xz8TqctJSIUi/+qBVNlWkdVdjZ3Zaio8tTRgeiJBOdNrpmpzjeIDBaS5WfTgJIZKRG5hZw9IGjLkUIwkIbSz04ZPMpnyjpyv97Jia6UXUSlYJCR0iJybpkpFTuImkus/P2W8DRnKREhOa7YVnaXtbV1xybn0pYUUXbt6onGNTrsgspuK6jJubSVZC9lqrK+yUQusNS0F9mlyUXe60dJwm0ckr/Vx+T1Fybccc5BBnoId0Imq7fPazMt3Anx6Mo4UOGscCcFtAhN1WvThVwPFu6EoIwj7RwrX+BuqHCnRmTv20rAs8JdmE5hDO9m0Xds62iwcG9gHriJcLh9sqtsv7xCq22JCluJXsKdOmuoI0a4o4wTwKLvsKzZZSdr/YSbgMJOQRXf3+ml/oTt51DhZsMx8xYymuq2VMa/86HCzYT5j+aQ4uhvRc4fXwQ3J47AXc/i+6gUkDlxDAts79hgMIBqol0c3Uy/jfrW5GKXzbnaAvSwO/I4zgr77BxmUb9B/cbksq42FFuT0wcpEVCwIRgFTAfa2sbksq42ICIXQZyiIDqdj7VfncllXe0/RRQ8UWtyWXHsssY5uyhJT3Ckz5jJpV1tW9mB7L8e3VlD+KOoo43JpcXRTnfSrf5Srp2/n+sbVR4Byxadmlze1WKZJSts5VjP1E6+oXp0kzVWnlqTCzvLulo9h4QtPg6CgT+yIk/oE7dxNwAAAABJRU5ErkJggg==)
which is not defined
(5) False
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFUAAAAXCAMAAABNu/MPAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bVb//FgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABH0lEQVRIS+1TXU/DMAx0+AxstGywFljG6GB1m///AznbiUCIKRtC4oHdQx01zuV8doiO+FsHNsvfur87ESr2DtBlEW/zXWl8lXbYJ9amyGYJw8ydvuzI7ZOssb47jJWn6y6z8gzlQY3EyZpoJfXKZqiMn/2+WokyK/sFDXWDs4itsCWt/WT7c9ZwsVUnwmWWZVzj7TLxs7/27vxzF4KUo/jaw6R1rCsltahfZY2trbqG4oNUsd8MfMsaWyhW1j6rSZ6yt9sLKGjV03katApxKKHogKoDYiv+ars/uGwVROVhWqlzC4rP6Iyr4J5RV4QfgL2tgK9NRxl2AtigwU+Ir3iYNzoPK51f9VaShjk8nmKSS4iP8rbP7kt5x/1/4MA7lEcYiQfDWD8AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
it decreases with increase in θ.
Verify:
cos60 = 1 – 2sin230 = 2cos230 – 1 = cos230 – sin230
Answer:
cos60 =,
=
2cos230 – 1 =
=
cos230 – sin230
=
=
Question 2.
Verify:
sin60 = 2sin30 cos30
Answer:
cos60 =,
2sin30 cos30 =
=
Question 3.
Verify:
sin60 =
Answer:
and
L.H.S.
R.H.S
⇒
⇒
Question 4.
Verify:
cos60 =
Answer:
cos60 =,
cos60 =
R.H.S
=
Question 5.
Verify:
cos90 = 4cos330 – 3cos30
Answer:
R.H.S
= 0
Question 6.
Evaluate:
Answer:
cos60 =,
=
=
Rationalise the denominator
=
Question 7.
Evaluate:
Answer:
cos60 =,
denominator =
=
Question 8.
Evaluate:
2sin230 cot30 – 3cos260 sec230
Answer:
=
⇒
Question 9.
Evaluate:
3cos230 + sec230 + 2cos0 + 3sin90 – tan260
Answer:
put all the respective values
=
Question 10.
In ΔABC, m∠B = 90, find the measure of the parts of the triangle other than the ones which are given below:
(1) m∠C = 45, AB = 5
(2) m∠A = 30, AC = 10
(3) AC = 6√2, BC = 3√6
(4) AB = 4, BC = 4
Answer:
(1) If ∠C = 45 and ∠B = 90
⇒ ∠A = 45 (angle sum property of a triangle)
∠A = ∠C
⇒ AB = BC (sides opposite to equal angles)
and since, AB = 5
⇒ BC = 5
and by Pythagoras theorem
AB2 + BC2 = AC2
⇒ 52 + 52 = AC
⇒ AC = 5√2
(2) if ∠A = 30 and ∠B = 90
⇒ ∠C = 60 (Angle sum property of triangle)
put the known values
also,
(3)
⇒ ∠C = 30
⇒ ∠A = 60 (by angle sum property)
also, by Pythagoras theorem
AB2 = AC2 – BC2
⇒ AB = 3√2
(4) Since ∠B = 90
⇒ AB2 + BC2 = AC2
substituting AC = 4 and BC = 4
we get
AC = 4√2
Also since, AB = BC
⇒ ∠A = ∠C (angles opposite to equal sides of the same triangle)
∵ ∠B = 0
⇒ ∠A = ∠C = 45 (Angle sum property)
Question 11.
In a rectangle ABCD, AB = 20, m∠BAC = 60, calculate the length of side and diagonals
and
.
Answer:
∵ ∠BAC = 60
we know that
since, diagonals of any rectangle are always equal
⇒ AC = BD
Now
by Pythagoras theorem
AB2 + BC2 = AC2
⇒ 202 + BC2 = 402
⇒ BC = 20√3
Question 12.
If θ is measure of an acute angle and cosθ = sinθ, find the value of 2tan2θ + sin2θ + 1.
Answer:
cos θ = sin θ
⇒ tan θ = 1
now ∵ θ is an acute angle and tan θ is 1.
⇒ θ = 45
now by substituting
we get
Question 13.
If α is measure of acute angle and 3sinα = 2cosα, prove that
Answer:
3sinα = 2cosα
L.H.S
by substituting
= 1
Question 14.
If A = 30 and B = 60, verify that
sin(A + B) = sinA cosB + cosA sinB,
Answer:
L.H.S
= Sin(A + B)
= sin(30 + 60)
= sin90
= 1
R.H.S
substituting the required values in
sinAcosB + cosAsinB
=
= 1
Question 15.
If A = 30 and B = 60, verify that
cos(A + B) = cosA cosB – sinA sinB
Answer:
L.H.S = cos(A + B)
⇒ cos(60 + 30)
⇒ cos90 = 0
R.H.S
cosA cosB – sinA sinB
⇒ cos60 cos30 – sin60 sin30
Question 16.
If sin(A – B) = sinA cosB – cosA sinB and cos(A – B) = cosA cosB + sinA sinB, find the values of sin15 and cos15.
Answer:
substituting A = 45 and B = 30 in
sin(A – B) = sinA cosB – cosA sinB
⇒ sin(45 – 30) = sin45 cos30 – cos45 sin30
Question 17.
State whether the following are true or false. Justify your answer:
(1) The value of sinθ increases as θ increases from 0 to 90.
(2) sinθ = cosθ for all value ofθ.
(3) cos(A + B) = cosA + cosB
(4) tanA is not defined for A = 90.
(5) The value of cot increases as θ increases from 0 to 90.
Answer:
(1) True
sin0 = 0
sin30 =
sin60 =
sin90 = 1
we can see its increasing with increase in the angle in the range 0–90.
(2) False
They are only equal at θ = 45, otherwise are not equal.
(3) False
Let A = B = 45
L.H.S
⇒ cos90 = 0
R.H.S
⇒ cos45 + cos45
(4) True
which is not defined
(5) False
it decreases with increase in θ.
Exercise 9.3
Question 1.Evaluate:
![](data:image/png;base64,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)
Answer:sin(90–θ ) = cosθ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 1
Question 2.Evaluate:
tan48 — cot42
Answer:cot(90–θ ) = tanθ
tan48 – cot42
= tan48 – cot(90–48)
= tan48 – tan48
= 0
Question 3.Evaluate:
cosec32 — sec58
Answer:sec(90–θ ) = cosecθ
cosec32 — sec58
= cosec32 — sec(90–32)
= cosec32 — cosec32
= 0
Question 4.Evaluate:
+ cos59 ⋅ cosec31
Answer:sin(90–θ ) = cosθ
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEYAAAAXCAMAAACBIsnkAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AABmADo6OgAAOjpmOmaQZgAAZjoAZjo6ZpDbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ/7Zm/9uQ/9u2//+2///bsoY2JgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAjUlEQVQ4T+2TyQ6AIAxEwX3f1///UfGClFZSo4kXeiTDTOYVhPDzG4GpaDnZbtmay3Cwbfa6so5I2aVZ0n5k2BCyJelBFMdGXUCyOQAwntlsmdQDfJ7ZmEXGwEBo2XR0GC6lejpszji8Kc3m81KvEGs2aOFwcbelsOzCvTexQhqV8NEiNrSM84e85icCB8mEB+N348+QAAAAAElFTkSuQmCC)
⇒ 2
Question 5.Evaluate:
sec70 sin20 — cos20 cosec70
Answer:sec(90–θ ) = cosecθ
cosec(90–θ ) = secθ
sec70 sin20 — cos20 cosec70
= sec(90–20)sin20 – cos20cosec(90–20)
= cosec20sin20 – cos20sec20
![](data:image/png;base64,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)
= 0
Question 6.Evaluate:
cos(40—
) — sin(50 +
) + ![](data:image/png;base64,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)
Answer:sin(90–θ ) = cosθ
cos(90–θ ) = sinθ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYkAAAAtCAMAAABs+WfDAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bfHcijAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFKUlEQVR4Xu1b61rbMAx1GNBt3FrYoGzQXQNb2zTv/3aTLMdxHDuW3bA2+ZIfgS84OraOLdknQojpmjwweWDyAMMDf++y7JrRLqLJG5iMQB9q093tk3g9ee6z+29gss/uHbOt7XuTiZc+aGmaPObBH7Bvm+zBQs/n9GA7y+CSC6T8mmWXv7i99JtECzktuTiTXOght2u57bVKE9tZxVG5PFsXy3e/m+MslzaH6u9+k5Jf4tZpcsiO7L3vuc7XNRMbdF7Lv14m7D7VJoXYLT5LJtwmex/NEAwWjxB5rtebDKIT3l5mcs+0gduG5nrNxAqXw25xHlgTYZNCrOaSA+E2OQTH9d1HjA7ll7M1+Xs7u3gWOXCyW2BusJjAtsgE3o3LXhMMk2JzuZZMeEz2Pcoh2Nst5kDApWZC0gGPjAvomWWn2m3kvA4mGCZ3t88Ul8hYy+QQPNd7H79lFz91DJILQ3rS9PTjvSiW4DeH21a4dORlHD6CJsslIAAT+cPEhOHo1w/ZKYR/FZ0cTKhkMdehJJQnRMgk5CO6FBOt1NP7fBuIwT8z8G2ACekslV6bSwYIau9iGSYbGdsyORDH9dpNTBFiVWdsx5pYoZtk8sBcHt7FMkziEIgJt8lehzgQY9sZJIE7mO/ymCBvdsZegcvoPIfbpnZ2tdcEwyQ6h87YbpMDcV6v3SweQcu4wq0TRG15ww1sI1gUnyCiX0mNowCFFho3L5sJhklcEirJO01yh1j+uLQO/Jw3cxpMwpWGJ9IBE/p4kFfK5Y09Lbr6gRLXNb6w/ejRZwKjiMSTmtpegAfxagKo3AzTxZETy+X5uiCBIE0TjsSDrcyegAk+OcwrG61FsuREkrjo3JNbp1PWACLxlKbmA2TLday+HbQRTDmFz5MTcYdYiWgpiyIWT25J/YDDY2ILu4QT3NLBT/ltRAmMpjLJkRPJkUpAMAJNazb5AA0llIOHwakLcHBMyD3x4gH8jpoLxCMtMKpAAxGHJSeSY6qJvbKEgpoPL2AknoLyAh4xE04Ni5Y4XNJ1OC21wFh5hiknkrycZURBZbYdX72AkXhKz/YCHjETzqRTyZH0U94rgTGnhO2TE21iSdzSU7RK2e1mCFSFMRMwgCeYgHIeOIRT/ejQv3iSf5MJ8qMSGNUc5cqJjbDtXxN+wEi8Zp5oAx7xmnBGJ8eaAC6kwKjjNjzgyIky7Owg5ehg52LfDxiLR873Ah4xE85VUUUTkrswTzQFRvUSR06Upz91JrC/wdTYfsB678SUL7sBh8YEiLX3ovwOn/hADCuQDi0w1vt7npwIiQLO2JQFOs4TXsBYPPzi3wE4OCYElDicPmF2AHkSRZxKYNTzG4MTS04EsfHkhmZ/1xnbC1ifsXl4KJj6AblMvEERa8vkvmWFXcczzwZAPU45YqtdWrdh319jAJtlTHYRK0dha/XCbzK5rLCJEauNVm//Ly1W9zYKsF06afDYqbDxawB1eA6VFXInXtr3gvTPBWl4e3+fqOpiQwV73ICH4ZnypSEZTTWAetZ5CvYgQ9b/PtFdsNdiImwyVFbIXRNjaucp2IPpWxMRUNiYNYCmSc3EVAOoJ5OvYM+oiw0pbDYTDJOhssIxzXX2WNwFe0ZdbGfBXkQNoFlqW3aUFbJ7PrqGexfstTN2yGSVtr1lhaNzMnNAexbsJdYAUvm/u6yQ2fFRNeuhYK9VebZ/WeGoXMwcTA8Fewk1gKGyQmbnR9Wsh4K9hBrAUFnhqFw8DWbyQJoH/gGWhdzKPUzSHAAAAABJRU5ErkJggg==)
= 1
Question 7.Evaluate:
+ ![](data:image/png;base64,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)
Answer:sin(90–θ ) = cosθ
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 1 + 1
= 2
Question 8.Evaluate:
cot12 ⋅ cot38 ⋅ cot52 ⋅ cot60 ⋅ cot78
Answer:cot(90–78).cot(90–52).cot52.
.cot78
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAAAvCAMAAAD5NPeuAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgBmZjoAZjo6ZjqQZmY6ZpDbZrbbZrb/kDoAkDo6kDpmkGY6kLyQkLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29u229v/2/+22////7Zm/9uQ/9u2/9vb//+2///bAqa9FgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADU0lEQVRoQ+1Z0WLaMAy0abusa1nZug3oWAldU1hHkuH//7hJsmObEBnSPS3CDwktOSsnnSzZKHUeZw+cPTA0DxQ6jKFxS/ExD0tJdAPX3ZetTOLlO5m8VT5Tyqyvtb6UJXnz/VmpfPSo6vlIFPPqBlI8R7kXGmIvZxR3Ddf8AmIvYfx5RZYrp+96cftTAmtVLzTqe/eJwrybaH0ro6z9eiVph2K2yaRIvcogvYuwoJXap/uwNW/mF89xv0qOEDFKPaNipswcKcOfImgj36s1Bdlg71LPr2SsbtSyXNsg14tM67EY3lDE/quF/KV/S81Bin/emblTDMwS3OiMvh1fJt4AqUCQMDkQN5vPp22n3gDpevXqPeNt3NupKoNrMZoC+WRJtLPwENYKGaCR66mq00K1s/SC8NEqkzu3AjLGzEk9eWqFjGbphLBWIhbHt1N2ll4QlvgKlQZVH9U8xsKnZy8ZfqKxm0BtaIhHyVOBKkcQKrzfwEbAzcJCeCuBBRncK7yMlRTkeDr6J8iLwG1rcvxUZR+WYRtbkItLEKFaR8qoMlTlDB6GOzRL8Ij/thvCWmmxiPeRnJUEpJt37o9cY3WHV6YJ3cV2ei7WagMrUIxpVE+bfUL4WRgIawUcnYUToj1OnJUEpEfAo1gF4qTwIDzb+ofur/na3unqiTmttiFdxG0ePaBonFdtnxmnWTN/bIWFRIfpJ5yr21cym3uIahPxhpn1ObSBcLPXkPn+lSjGnhgDSVlB0bg2M67B++7dt8JAmIAnpI453qgc2Dub7n12E7uq+5YoEXEO4nP80Ap50FqAlzgh4s75HZA+SrexosuB1B1VF+tQzpo8tl/s5TgHYa3kGGvrrgJnK73YWSs8pBdxNPr0Ay71V1qlrN598uJchf64pQbGRQb+MVXmaQkPA8yxh1mWjVQOIawVLCW2MFB3YoDVMSsJSC/mK6zIUGjHvyewmcVMx9OqeMGilvXy0UtSKSj1+Lda+x0QzRKtlG0IZ6W+h3WIzgRdJgbinJUUpBfzoT18ctM/NOInNP1Do2z5SPwNJURSzmlbS71ifjxq8W5tZIaZ0x2s9pp+MazDrlIQ5WZdl3OsHAe31fSLibtv+sUwdkR90y+N+Jmv9cBfQ/xytAZ4JeMAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Question 9.Evaluate:
+ √3 (tan10 tan30 tcm40 tan50 tan80–
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
![](data:image/png;base64,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)
2
Question 10.Evaluate:
![](data:image/png;base64,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)
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALQAAAAuCAMAAAC27sMlAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpDbZgAAZgBmZjoAZjo6ZjqQZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6ttvbttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///b8AEBqgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC10lEQVRoQ+1Z7VbbMAy1yoAMaOm6r3TrCGMZhaZZ/P5PN0kOdmgdy23TLOxUP0JOsK+vlWvZuVXqFMPIQAFfm0T0cgZwfkeP9ALg5iGaZStQAzIItvzIw8bExlgZfFHl9OwXck4vVmVKd3HRCuQgQ0DlDOLH2gDKLvFBTtkvRjjxDSZx9LmVA3J3ge7r8UMuky7nAKPJqgAkSJffCUxeQJlqRhjVlCYRDBGIe0fMXiZNL19/v1ipdYIE18n1ncmvyRLypQZEmq6hEIEcpDB7mXQ1vUWuN5Y0M8dHFDwPQ9pcQyECOciDSat7uP5pGTJNHh5Dp/Q3lrQI5CAPJ60er+AcVVvLw5HWKcv4RR6ipiUgxpJRcPnLCxGxlglibZHOLo0g6oVYSyaYpTAQVRJBY4wukyY5q8wtRJvpnERcIFVT915vPR7qMpCykGF9RJBOcBOZYaaZFl94Ia7fY3XWGd5R4ZDXIfaSgBykQJo2hmCU8wRgTMUDazRfqinArcqAg0SBexS1OBioAdmOpRfIAd59loY7/f+Uga4zoNNXJ9mu4Y+DdyJ9nLxuo/aV6bpOYhkUq7c89b5Ie5mYPSQqbH/v7KMgTCM+0dj2dQa9/eXk7dSir0z/P/LYKb2Nxn1lel9+3n5HJJ07eXdKmT5CjrUj6rl0MnVTGYzDVH0Sj8GW9VAcJlXEfEf6dfOvHCas4ORb4Nd27SR66YnGUM8Ok/6G1kA2+qHKtHXjFo2hvh0m84FsnURvokVjKNasEYF4eDYqgpFbr4JfsT+G4zD9eSaG93XBKxfjgCc+EIepXABpovrA6SVLIOwADMJhenpmObiCt0xa5SEbQ705TGwS5U7zBbsuvpCNod4cJo2/kTT3cGs5b/EeksOEmzK/+Npklv3Djs9Te8FhdX1kRWjaV0rRx99rkM475XBlJF2SozeJPzh1zmQHwIp/A3xrke9/wntrUx0Y379tpniH4+84GgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 11.Prove the following:
sin48 sec42 + cos48 cosec42 = 2
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
sin48 sec42 + cos48 cosec42
= sin48sec(90–48) + cos48cosec(90–48)
= sin48cosec48 + cos48sec48
= 1 + 1
= 2 = R.H.S
Question 12.Prove the following:
— 2cos70 cosec20 = 0
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 1 + 1–2
= 0 = R.H.S
Question 13.Prove the following:
![](data:image/png;base64,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)
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0
Question 14.Prove the following:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 15.Express the following in terms of trigonometric ratios of angles having measure between 0 and 45:
(1) sin85 + cosec85
(2) cos89 + cosec87
(3) sec81 + cosec54
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
(1) sin85 + cosec85
sin(90–5) + cosec(90–5)
= cos5 + sec5
(2) cos89 + cosec87
cos(90–1) + cosec(90–3)
= sin1 + sec3
(3) sec81 + cosec54
sec(90–9) + cosec(90–36)
= cosec9 + sec36
Question 16.
Answer:(a) cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
A + B + C = 180 (Angle sum property of triangle)
⇒ A + C = 180–B
dividing both sides by 2
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPYAAAArCAMAAAB8bIULAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAA///b//+22///2/+2tv///9vbtv/b/9u2/9uQttv/kNv//7Zm27aQ27ZmkLbbkLyQZrb/kLaQZrbb25Bm25A6tpA6ZpC2ZpCQOpDbtmZmOpC2tmY6kGZmtmYAkGY6kGYAOma2ZmY6OmaQkDo6AGa2kDoAZjo6OjqQZjoAOjpmOjoAADqQADpmZgBmADo6ZgA6ZgAAOgA6OgAAAABmAAA6AAAA5escxgAAAAF0Uk5TAEDm2GYAAANwSURBVHja7Zptf5MwEMCT4rQM5sTKFK0oKnWbnbMy7vt/NH+5QClE8nS07EXzorQb5f73kLtLUsbO4zwsB99sA4vbru4A6u/BbJAgRl1MBhBDHalSksFfUvgU8Bv1zlONxa5kjN9ASbfg50txyT9UmVHtsBLy+I+IhcVUDpTird1dovLbgAqwRm0Xv1+pUa6onXZ+XmfTqO32nFbtkgqQlu0lfbowqM033S3qBKCId1L75deHCyJA+DdqlQirxKC2CK4uGShW8hfvmNO2l1SAXFo7vA2aANr/A9rRBXZPbb6ZIMxzx9yEjDxHJAJAa+0UFRyaTxfkk7jb1dkHKY0EkEr3Lf5EWMMSQ0rLD0vXYkd2d7oNfNRuzO8N0H4xRvGYILVqx5A1BQyNsCV2De7cPW97A8R11D2MLXaDqFHU5pt6xfbtSkxtW9wfINuVHDISQNoEC8A2wEtfb7VG8HUF8Pg+aJoXYpSnrpOzaU4fV4wCYNmH6yPu6MuAyQGUyexafmjYVPG+ANQozWklrBMv1nUP2IPwLwC3l3QAXWlUKpbr3DSnFCvxaX3NOKog4n65sbbmOIAu251AbRvxso7hK95uj9UDCO+7KFnL7ou//gVQiCdC8q6CwhrbsEY38VmJl4sBTHDocvs53wc4NDG+55sykJ1sWN1HLG3uPr7aVuKlt4XaMrf3Wn8jgKi76vKhMwEatXnJplf7f2sXS/H5UyS2TFq17Qvb0O5pnWjktk1hKxfch5O3DeIxfT+83ZVsoLZiSRPAUG1+9bMCGJV75CC3Ei9ioA1yl7mtCXIxuXTmPnKQW4nHT01Ky6ZJabFMJ0O5pypgNuKlizHbTVDAMHY+vqkytvxWtY9sc8op2hVL8VdVETRLSodWXQOwhjoR9bN4sYMkFFNMzIbsVM2plfiwur+W75Z3APanD37dsTZ3cGyUo+e1FBlA+QFowymHFVvudOacYeHZh/IF0K3z832KOdYCzn2bYQDlC2DcldEmvRk2lfpQvgDGCqnNGTNsIfahvAEMO7bqQQkdmrRh3IPyB9Dvz+vPHeY4HuhBEQB0pzH6RDnLYdAhFAVAZ++8DI7rbB93d1AkgPGTVpx4cTZq9TkOeg+giABjx+PoCJ5njl+bSrwRiggw9iuKZiE5ZpR5fsTRQU0GcB7n8dzHP8Qbom8bgdkHAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
(2) A + B + C = 180
⇒ B + C = 180–A
dividing by 2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 17.If A + B = 90, prove that ![](data:image/png;base64,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)
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
![](data:image/png;base64,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)
∵ A + B = 90
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∵ ![](data:image/png;base64,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)
= secA
Question 18.If 3 θ is the measure of an acute angle and sin30 = cos(θ — 26), then find the value of θ.
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
sin30 = cos(90–30)
⇒ cos(90–30) = cos(θ –26)
⇒ 90–30 = θ–26
⇒ θ = 86
Question 19.If 0 < θ < 90, θ, sinθ = cos30, then obtain the value of 2tan2θ — 1.
Answer:cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
sinθ = cos30
![](data:image/png;base64,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)
⇒ θ = 60
Now, 2tan2θ – 1
= 2tan260 – 1
= 2(√3)2 – 1
= 5
Question 20.If tanA = cotB, prove that A + B = 90, where A and B are measures of acute angles.
Answer:tan(90–θ ) = cotθ
tanA = cotB
⇒ tanA = tan(90–B)
⇒ A = 90–B
⇒ A + B = 90
Question 21.If sec2A = cosec(A — 42), where 2A is the measure of an acute angle, find the value of A.
Answer:sec(90–θ) = cosecθ
sec2A = cosec(A–42)
∴ sec2A = sec(90–(A–42))
⇒ 2A = 90 – A + 42
⇒ A = 44
Question 22.If 0 < θ < 90 and secθ = cosec60, find the value of 2cos2θ — 1.
Answer:secθ = cosec60
⇒ secθ =![](data:image/png;base64,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)
⇒ θ = 30
Now,
2cos2
— 1
= 2cos230— 1
Evaluate:
Answer:
sin(90–θ ) = cosθ
= 1
Question 2.
Evaluate:
tan48 — cot42
Answer:
cot(90–θ ) = tanθ
tan48 – cot42
= tan48 – cot(90–48)
= tan48 – tan48
= 0
Question 3.
Evaluate:
cosec32 — sec58
Answer:
sec(90–θ ) = cosecθ
cosec32 — sec58
= cosec32 — sec(90–32)
= cosec32 — cosec32
= 0
Question 4.
Evaluate: + cos59 ⋅ cosec31
Answer:
sin(90–θ ) = cosθ
⇒
⇒ 2
Question 5.
Evaluate:
sec70 sin20 — cos20 cosec70
Answer:
sec(90–θ ) = cosecθ
cosec(90–θ ) = secθ
sec70 sin20 — cos20 cosec70
= sec(90–20)sin20 – cos20cosec(90–20)
= cosec20sin20 – cos20sec20
= 0
Question 6.
Evaluate:
cos(40—) — sin(50 +
) +
Answer:
sin(90–θ ) = cosθ
cos(90–θ ) = sinθ
= 1
Question 7.
Evaluate: +
Answer:
sin(90–θ ) = cosθ
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
= 1 + 1
= 2
Question 8.
Evaluate:
cot12 ⋅ cot38 ⋅ cot52 ⋅ cot60 ⋅ cot78
Answer:
cot(90–78).cot(90–52).cot52..cot78
Question 9.
Evaluate: + √3 (tan10 tan30 tcm40 tan50 tan80–
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
2
Question 10.
Evaluate:
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
Question 11.
Prove the following:
sin48 sec42 + cos48 cosec42 = 2
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
sin48 sec42 + cos48 cosec42
= sin48sec(90–48) + cos48cosec(90–48)
= sin48cosec48 + cos48sec48
= 1 + 1
= 2 = R.H.S
Question 12.
Prove the following: — 2cos70 cosec20 = 0
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
= 1 + 1–2
= 0 = R.H.S
Question 13.
Prove the following:
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S.
= 0
Question 14.
Prove the following:
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
Question 15.
Express the following in terms of trigonometric ratios of angles having measure between 0 and 45:
(1) sin85 + cosec85
(2) cos89 + cosec87
(3) sec81 + cosec54
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
(1) sin85 + cosec85
sin(90–5) + cosec(90–5)
= cos5 + sec5
(2) cos89 + cosec87
cos(90–1) + cosec(90–3)
= sin1 + sec3
(3) sec81 + cosec54
sec(90–9) + cosec(90–36)
= cosec9 + sec36
Question 16.
Answer:
(a) cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
A + B + C = 180 (Angle sum property of triangle)
⇒ A + C = 180–B
dividing both sides by 2
(2) A + B + C = 180
⇒ B + C = 180–A
dividing by 2
Question 17.
If A + B = 90, prove that
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
L.H.S
∵ A + B = 90
∵
= secA
Question 18.
If 3 θ is the measure of an acute angle and sin30 = cos(θ — 26), then find the value of θ.
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
sin30 = cos(90–30)
⇒ cos(90–30) = cos(θ –26)
⇒ 90–30 = θ–26
⇒ θ = 86
Question 19.
If 0 < θ < 90, θ, sinθ = cos30, then obtain the value of 2tan2θ — 1.
Answer:
cos(90–θ ) = sinθ
tan(90–θ ) = cotθ
sin(90–θ ) = cosθ
cot(90–θ ) = tanθ
cosec(90–θ ) = secθ
sec(90–θ) = cosecθ
sinθ = cos30
⇒ θ = 60
Now, 2tan2θ – 1
= 2tan260 – 1
= 2(√3)2 – 1
= 5
Question 20.
If tanA = cotB, prove that A + B = 90, where A and B are measures of acute angles.
Answer:
tan(90–θ ) = cotθ
tanA = cotB
⇒ tanA = tan(90–B)
⇒ A = 90–B
⇒ A + B = 90
Question 21.
If sec2A = cosec(A — 42), where 2A is the measure of an acute angle, find the value of A.
Answer:
sec(90–θ) = cosecθ
sec2A = cosec(A–42)
∴ sec2A = sec(90–(A–42))
⇒ 2A = 90 – A + 42
⇒ A = 44
Question 22.
If 0 < θ < 90 and secθ = cosec60, find the value of 2cos2θ — 1.
Answer:
secθ = cosec60
⇒ secθ =
⇒ θ = 30
Now,
2cos2— 1
= 2cos230— 1
Exercise 9
Question 1.Prove the following by using trigonometric identities:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAA0CAYAAAD4xrjIAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAYASURBVHja7Zy/b9tGFMcf4zV25iQ6bkY7xjxp65jQZ8Qp0IF1ZUJL3BKBYBCI1YBD0FDeBKR/gIsMTjJ66NYClZ2pS5ruRYFEf0Bs/wmpVR7JoyiJEn9IrUjxDV9AFinyyPvw3bt3XxMODw8BhSqS8CagEFoUCqFFoRBaFEKL+m87CEA6AbgGoK3g/UBoc6++A2xzR92+R+UXALTXsO2beF8Q2lyryciBLMNfEsAVQovQFkq7FF4htAgtQovQohBahBaF0CK0CC1COxd59UdYwRokQlsIaC3LWN0i0HWg/SSB8kGzrNvYIQhtrqHdZ7U91mrf4Z91Ci+hws6MTuc6dgpCW4ic1jbZBgHlHXYKQlscaO3GTQq0m7ZTBjkxV19aus4BkAxDu6FS+BlA6d1tPqUAJ9cQ2hw04mnzrlJj5l6azrSa2mesKv/Ec2IAck7VfYN/v0yd027Q+971hUR3j5ceyhiT0MIbyE0hOmN7SfNZfkEO5JQQ8jvTH2/x70y9tiMBuXQikYLDZ7GVxCS08Abuq6RFG+37AsgEUZnKIP8tJnGD9AJ6RN1vJYC+9MNrnpXEJLSwxvHw/0glT4j66AnPSS1Lu62q5tfTfmNZjVuKAydQ/eX4RA4u46DlcCugnOKEptiTz6kRaTDJGU/+h7e7n6XJ+5y428OTJadRx8P5mvwxHD2j5ETl75389WJ0v7ZOt51j9AkzDxDaEkIrJjmUwB8+UP/I9N4LzbDWwturhPwaAEeUd97MdgCu7UROSiT3GHKVHTWbmkKpbme9iCDKKt+8Dj0srnjEdmC+ZKa9gdAuVsMVnWmaXu1JBa1dpw/cnNGf5PhD74Uo/ruTIJAuCGseeNvr684Xb3mk26j/8GVwUgVeU9X8ln/udIzrbmSlu6+y3gwvykr9sdm081DxczsX+CEORoT2f4mQxxF9NCbBz8zQcrhYBd6Ec0MPOPIcyGa3bpvrfPvo6lUAtn8SMTHiEIvUwv2O1p9luRGiXVEXwR8yF9qRPDfqqTdNreJAe6aZZmX4qcfaZ2HTAzGhoXp7OzJS+dtHI6YP1ZkYovnfvq/giqcWXxmtzyflvUkUnJew0/DDwo/HI7qb50akBhOe+qsktc9kQ1y5lQtoxYSG1u0HUQcS26OGee8kzm994ENpwye3fEGUP3nem+UCgonWSHWAL0p4MG91k9R506QHSYa4sisXOW0Q0dzJzvBBeU11kN+qbwzLWh0/ySDiiajK4VWr8pGXeyq9hmXdygpteAQIRdnzuKoD5rRLnNP6ueh7Dkhlc+dQM4wb4sloqvS7mm7uMAKno5OujmWsqjzn9CMeTw9Uqj4TFYcBeFJfLCRkgtYfAYJVMYDzNMcrM7RF8y6nqh74693+jFxIuhJADrYrPT9XXfHrp0HEc3NcB25C9R8d8Nfcfeq1utOI91mAabfYHQ6oKHfxZT7+N3+IUuXGJYW2SN7lJCahyB8+1tkWX0oToVyuqkd1014PQyTyVXe7XP0lvN2FlukPTZN9EexH6Nukw3iUuL9ABvgojhU+XxpoKdDfygZtkbzLSUxCmIeVTMvgXcaOLBu0GbzLefMtY0cmyLGW6QVwabzLefUtI5hTVMQXwPWnAJXGu5xn3zLCOUVZXwC3KM+uV7tW2lHtTOtdntW3jNAuWGn+sXCRZTWvdj0+yUrrXZ7Vt4zQlgTaOP/y+D7DXg7L0NbcBR5QeiFDkOS3P5V3eVbfMkK75NDG+ZfD+0zyMPO6+CaBrpfGCPtmvKl+apSdwbeM0C45tHH+5cGwn8DDPIf3I8zDt4zQ5hzaWTy7cf5lfr7ASxzjYZ4HtFl9ywhtwaCdxbMb518O7xPnYZ4HtFl9ywhtidKDOP9yeJ8kHuZZoZ2XbxmhXWJo4/zLkNLDPC9oZ/UtI7TLDG2Mf5kvBAibZ5yHebSdWZZa5+VbRmgXeXMyvAAubfUgzr88vM9kDzOXazv0llmpztjDtEP5vHzLCO0CleUFcFk8u3H+ZQHUNA+z2Mf1HM/gXZ6HbxmhRaEQWhRCi0IhtCjUdP0L1nfaYzyz6jwAAAAASUVORK5CYII=)
Answer:1 + cot2θ = cosec2θ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 1
Question 2.Prove the following by using trigonometric identities:
2sin2θ + 4sec2θ + 5cot2θ + 2cos2θ — 4tan2θ — 5cosec2θ = 1
Answer:L.H.S
2sin2θ + 4sec2θ + 5cot2θ + 2cos2θ — 4tan2θ — 5cosec2θ
= (2sin2θ + 2cos2θ) + ( 4sec2θ – 4tan2
) + (5cot2θ — 5cosec2θ)
= 2 + 4 – 5
= 1
Question 3.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 4.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHcAAAAqCAMAAAC+9zSRAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZmY6ZrbbZrb/kDoAkDo6kGYAkGY6kLyQkNv/tmYAtmY6tmZmttvbtv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///bVE7vrAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB+klEQVRYR+1Xa1PCMBBMilBFkaIWVEBAsbTk//8/71Kb5tXmMTjDjM2HDtDsbW7vErKEDKNfAfZG6ayIVikWzpaTosom3rwnmstzXfDydm0PfUrgBX+o4ZTo0iuN1w5v5xuRm1ebMWhcphoNW3bxaqu3w8WkLYUx2rOvOygm5Edp/pniJ9AJJp2zuSvfakVpMisQ6YZr+WIx2AY1LdPpmhwgRM1bP6Vh5MuW44K9gjZcGhfcpvMvkKPnkCkqQamLlytS3gvefng/L8Y681ZW8t3UK4Ehd9uWTncYrl12B1wVru4rdnxKsUA1GoHe9SXQGjd7nbcDbqmvtmDCG/Kc+fTzMQVt1Hw74BovT1oDHvj+HUEmvX2FpeUkXvA2FPbQxzs8qmeUmO9+/A0LDOeVto1Afk2AMn0h1QLy5UgXXEphS5OcwDaefWc0L7HK2MrAVy1o8qidBiZvtQLIA6YLSCdcDzd8HxQYFPBTwH6s+2EvPkv8xfzVh4uv2DvgVensver/MTHy0h8JE5q6Lv3NxECv0MACPYNZ6kCvIJaru5HmhePS39VrnrBQz9DSRXoFJV9/zyCVP84rmDp7eQbBG+sV+ngtl37zwIv0Ciqvt2eQ2inOKxj1Db30Q4AYr6Dw+nsGkW+sVxDnRphnaHljvUITIcwzCN5r9go/JShQNHeQhI0AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 5.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 6.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 8.Prove the following by using trigonometric identities:
(sinθ + cosecθ)2 + (cosθ + secθ)2 = 7 + tan2θ + cot2θ.
Answer:L.H.S
(sinθ + cosecθ)2 + (cosθ + secθ)2
= sin2θ + 2sinθcosecθ + cosec2θ + cos2θ + 2cosθsecθ + sec2θ
= sin2θ + 2 + cosec2θ + cos2θ + 2 + sec2θ
= sin2θ + cos2θ + sec2θ + cosec2θ + 2 + 2
= 1 + 2 + 2 + (1 + tan2θ) + (1 + cot2θ)
= 7 + tan2θ + cot2θ
Question 9.Prove the following by using trigonometric identities:
2sec2θ — sec4θ — 2cosec2θ + cosec4θ = cot4θ — tan4θ.
Answer:2sec2θ — sec4θ — 2cosec2θ + cosec4θ
= 2(1 + tan2θ) – (1 + tan2θ)2 – 2(1 + cot2θ) + (1 + cot2θ)2
open all the brackets and cancel terms
= cot4θ – tan4θ
Question 10.Prove the following by using trigonometric identities:
(sinθ — secθ)2 + (cosθ — cosecθ)2 = (1 — secθ ⋅ cosecθ)2.
Answer:L.H.S
![](data:image/png;base64,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)
= sin2θ – 2sinθsecθ + sec2θ + cos2θ – 2cosθcosecθ + cosec2θ
= (sin2θ + cos2θ) – (2sinθsecθ + 2cosθcosecθ) + cosec2θ + sec2θ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 11.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 12.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPMAAAAXCAMAAADp/YmZAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZpDbZrbbZrb/kDoAkGY6kLyQkLbbkNv/tmYAtmY6tmZmtpBmttv/tv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///b6AkJ8gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACjUlEQVRYR+2XaUPbMAyG7XIFxtlxJUA7jqwMmtb//9dNluzEh+Y4bWBf6g9pKYkevbrsCLFbuwhsH4E/11JebG9mAwsbkdXL2dsGLO+R9c8nsZjMfDP1+Wva7leR+8HV5ee2kvH55thoVg+Qc22z+VGmLKuxyfngK+PX+02Qp4GRqI0hVR19rqZHXhg4U6oamZwLFss9KuzVtTTfHP9UFSSqzWUsYmHbealrHC9C1If/rqEUOQIzQWt9seRcMMSGCvP8tc7QbKQwHtTtBJtrnU2B0UrEKEnO0Wx9acmZYOsb5qRf8y8Ja+9NLU5wSi+lLH8XOK+X+k+USVrWUyzcrn6jMJmo8GRGcwM7wwQI+vMMhqPxpSPngm0NZmqm23XfqLn+1hSnM1HLEiTqaDiabRbnVEbMckomjnasuSnuxWpaAhI+K50eMtCRCemB59orXO6ekiSDgaCfu9sxTeZiZxEqIyekJLFYb2NotoYwiMgN+ywX7BZ0EG02RpxmU8ZW2BpHdhtuqzky5+B8Mgu2EPrEa6yZBccB3yjP6v2mwErGeAeavbYaLc++ZmSEmhnweLWt+9mWdayZZK6h9bC2R+rnjDznggfObYotXrp+DvIsaryHNoHwf06hDZvbtldU1e6E0b6ZCW77DpzBR/wVn0mKK/H8CJfVLY4RKnBvhoFOfQ6j3zL2Z5Ycz+1a3gv1PAMkwI1w8MV1ORPcJkQ9QIfK/bsezbArwh4JW+PFx1SWje5qPS590XCkm1ySoYxzGEtm9mc4CBw8gckFQPE0T764KxOcOjXwe8yQXxNpTp5XhjDYe5NgjR7pvSqGf9d7VUTuAYPoEd6f2WD3v8Z+EbkXvHUd7QzsIvBfI/AXK4dXej/+dDYAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Question 13.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 14.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:L.H.S
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∵ tanθcotθ = 1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 15.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
∵ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWIAAAAtCAMAAACj+5FFAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZpCQZpC2ZpDbZra2ZrbbZrb/kDoAkDo6kGYAkLbbkNvbkNv/tmYAtmY6tmZmtpBmttvbttv/tv+2tv/btv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///b093zRgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD8UlEQVRoQ+1aaWPaMAx1KCM76AZbd7R0XWl3ka5rOP7/b5vkixx2LBsnayD+UNLEsqQXIZyXx9gwBgQGBJ4tArvFi6eA4ALNAjz13yRPRrf0LNYv5WQ/M7qDY5y5/JjO6Hnl6n74mdEd9GTm+iJJRleM4ef0JwS9uYYTb6EfwJnRZ8xCTWHbD78bOkV1pbsExtmKOcx6glN4mOv0km3mV4x/LgAQ7Ju7r9By8cxjiuDLKYxlE5YhZsZRW4mpKm40Cw+9L5ZL+fu1nGC5AqLbObSC9fSJ4ZXdAk6rKfAf3goA3ThqKymIm836AlRwnBxQGOKT/71L3vzQZwA3NUUAz0HHscQ2wIfouIaVZBWXzIIj7a1hGRiB359XyXgFkHH8zlZ7iDN5xpitYSUJcaNZb5EjB26oPbB9TCe6KgsH77Fc88TcKaxVvG00I0fa24nqey8eDrDR4tca26/uCPog5217O5edopJzfSXZi5vNeoscPfAsuWS7+1sozhnsKORGYnMBMOZ45QF2bXKKxHA7t+wpaivBHZux+2+ieVvN6KH2duZDmoxvsAOnSYK74c01HJxjvcKV0Sd5ML7B3gx3AD8sGFdXwl/O0RenWW+RGwIfEHgGCAzsVOs3YWCnWof4xNmpQHzjkVqBARy9WTxS6+ihcidYpUiERTxSyx3Bac4IJ7VqrJYmuU71wFJB8Uit0yzRctbGRhGP1BogtiAQj9QaILYhEI/UQg+771PbW7R2bkF2ji9PxejcufRbjMGYZTxSC5JcvAsRjRyC/vq1IuP/g3MZ+D6GQzIh2e4WM9K84qRQ1kPbKelJl87V10aJC7T8xTt9X4Pc+q7dvpIn62HQ8mTidXUk5/hm2jHsMbgsD76uX/b4rOTJehi0PCLjWM4JEFtj8Mk7bK5WM7hpD6KUR0t+lCrIpOURHcLq3FNHVIeYoicK6VIhIKu766Y9iFIeLfnRqiCjlocLWmzOfXVENYhpeiIeQ/tDZemmPYhSnjqBYtTy8NVszn11RDWIaXoinVG7MEtlmZv2+CvVPhUpT6iWh6dndU7UESE0pudfop6oI4hlIblpj18a4mZNDlHLU6xig4mfjqhaxT4xtFvBfNtk1ZHBxaKWR6ulHJocopan2IuNnIuPjqgZYr68yLMce0e9WP6ou2kPopRHb8T2qiCTlkeganPuqyOqQkzTE+0ldu1WsorGTXsQpTxK8rNXBZm0POV9cdU53xB46IhqP3ckPVFnj3fqActNe/AZbikPan9KqiCTlqfydFcx8dUR1ffFHjG0W8K4elcb8FImh3AUsSDprIgHpi3WLWtap3PKthd8cRfIDz4GBKIg8A9qP9ThgGHOUwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Question 16.Prove the following by using trigonometric identities:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKEAAAAqCAMAAADs+MufAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bf9mRtgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACSUlEQVRYR+1Ya1PCMBBMUKTyEFAeIqiAAi39///PzYVWHrWX9IIDM+RDCkOyt9m7pGGVurV/VyCdaN1Zi8IGgNiPv9HD/a/pqLFOeg0JwwAQB+GPGG5qU6Woq9wCQJTFnj0gw3F0oKsn1wAQFDEZa13rrDcaWTbdV6Q7SiFD+G3b6/Ks4j4AsBLzbM1/ET0gSoOko4d1+grBSK84ak7VElwtvO3LWxwNVNIbYiqeo7sFJllEd4jyACRT3MoZEs0u1NPUeIaUS7SZGWqWmSG6QzAavOvmpxmy0xAMTYgtbWMHDbNCsE/qd4jOEFyW1Pejri+OGToX0SFDO80iOkOwDJVaRcA91FBR8raoL6YVaIgZhOgKwRU6SpCwjhgu6TxE4TMtKwSzP+wyTVEToisEx9BswT5WTCc2dWanmELEO8XhsFnqgUo/ppjaxV6mpe4QnSHKKSbjSOu2wcVZSJ3Zg2CW9HXtiVPQ/I4DtP5mqg+TzYs8Q/SAcAlzG3PlCqyeJReUo8WnI/b8KZfrFAAFzx8Z7jkIzzBuz5eXzRDqXA/Dmb2toAluzuGzzGmYs/77Q16khUsUA3AM3feIHXlWDa80yxeloS+ZwvHnyXLVrVtkGXgwLHYcTgDSCe5E+v6lgoC+lkF404Ij7WsZhDctOIZCy0A4fY+d2DIQmxYu/1MkloHYtHDKJQZVtgzEpgXDMIBlIDQtOAlDWAYy08KPYcbX1zKQmBYcQ7llkFsMFU0LjiHMQqFlkFsMVU0LlqLUMpCbFizFSxnwA2oAWFXlDd66AAAAAElFTkSuQmCC)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALYAAAAtCAMAAAA0j2G2AAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///btSi4tQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC3ElEQVRoQ+2Y7VobIRCFIa26rR81TauxmtjqNjb7df+X14EFFtgBNjhpo0/4wfooc3iZHVg8jB3bm8/AnwXnV1SrIBWLQbVfH9hmtqLhJhVLItWfFHZ3D5nfJsdHB2gxCi17oorfevOW1/0vuuXZtpmfvQ5biZFoRbE3urQrUSyys1q39FcZXZYWQ7VelxA3ujQbcn0KBVIXLuZu2EYM1crBbu44n11tKw5FIrrfhTxBKugqSQrvFfp2rkpGzYFg13D4zCBEPC+eYJyStsRwrQzsbnm67X5APmU66+J8xUpYQDvn0CzsfsKhjbHr4oY181vQgOfywzOsV0kPYr2Kr5WBLbNYXxhsye4ktp+U8xS2fP/Q1mKgSIKRHrBwrQxs9sjPf4k4lW3A9uqhlYeInaF1vw5o1j7VUf1T9lraxibKNmObz/zkOYI9sbZd7D5ISdulhe2TnGxDzEsBYsFsM/n2W6jaaG0j2dbSVhyqlYEtylqChbFLeW7DHoti6yoSG7FPgpG24lCtHGyx7xeQbfmVlJ23JSHR4ivpbFNR66OPKr9h3c8VaFzDSSIToaQtLFQrA7u5Kzi/FHPAeSc7sdtdxmbBZ188beTchhP/5EFUNKiIK4yWdiIxrQzsYwhVBl6+EV2YqYCm6EAVebtffjV2urhNmYd2TH35VL5BbEjCEZu2EqQaerMRf3gf2UaXZy5z//iHyOs77Gy/8yIR7+Xgz22xJce20sFjd/dwEeMfv7u1T4Sd9JaSA/ZwoiYlx96S52mRm09JpCkDxt6Sh71382kK5WhM0ltKDsiadmJQyGay/pfHPS0y82kiqDMsaDMN3lLA0yIzn3KwgzbT4C0FPC0682l37rDNZPlUuKc1NrJ2nz43Imwz2aWLelr/s7YjNpPjLWGeFpX5lJHyiM1kvKWQp0VlPmVgg88dspmMt2SMJ8/TojKfcrBZ2GbS3lLQ0zqaT1kZ31fQX82CZBfoNO3zAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 17.Prove the following by using trigonometric identities:
sin4θ – cos4θ = sin2θ – cos2θ = 2sin2θ – 1 = 1 – 2 cos2θ.
Answer:![](data:image/png;base64,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)
. . .(1)
Now, ![](data:image/png;base64,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)
Also, ![](data:image/png;base64,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)
By (1)
![](data:image/png;base64,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)
Question 18.Prove the following by using trigonometric identities:
tan2A — tan2B = ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMwAAAAtCAMAAADV5HTnAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv/btv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2/9vb//+2///bQ6Bk9AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADkUlEQVRoQ+1Za3fTMAy1yx4B1o5lDOi6sWYbDEIHSUn+/19Dkh9xEtc2dcJhPcmHpGezdHUl+ZEbxqbrMDPw84rz839JzQtYf5l/3y+g6vKObWbrjnG+eHK7GxGwXr37tR8Xsipfd8mw8u21y+GIgPXqQiI/f+iF1Yup4N04c2Ferzhes3NMjIVg42kgQMI8kWVQgMUr0WPbKy5/ubLaI7NRU6ZKT8H9I4c7Y7lCsfgaCpCxgusWF4D1itBZuXjKA8h0g8v19BeOqpS8OkozGCBj2cdEdZUALBPdNy4y2xtqoYJDm+HtR0KrWAG3QjhQZCg3TSv1SjMYIKvef9N9JgCLZjFykMHurG8h6RRKmZytWQ60qhTniUHm9+2xaNpM1NvWZUMBsvyUNSETYBiZKgXi5VyTIUaqxCJisQKcyFU52zlpBgS8NvqKAI1yGDRpZaLVCVsRrnt+9hWfsjJAhvgZF7VZnUkDRSbreRoMEHOrJiC0ApIJSxRjmzccO8hDRi0AwndEmwUA5pQntWj9FRnGnhNIvY9MvRLeY+cM+nADVpfYNHqrIMCwxQVLStn2kZFLc7cHjRoNBVhQ7Wl3oyc2fdN1LO+dsnQMZbKEXRXsKBN06y4AYtPMOM2kgH0GRsUAyrirVHSCBJQbcv05gQ48+mRv9O0N/HeBhYGVmG64KpsrgDzOHC3JQcAJIA4Q4aE0+CA2EtCxv+2Ywv4/B57N/I5CR2jAyEOsBW/cU7MbcP/XC3viRnyf2RMwtMDTuCkDUwZebgb0ofYAfrzcKkyRTxk4jAx4pWArzf2sxs5Yoz23BE4PbFuxbmTKsaMN8C9P4y2B02/WvDQYMqXfbOwRWns2BE4/prSCgYZM6TcbYEQJX2Vm+OYMzzkKZVLvRNdKe24JnIQZYgUv84ZMOUCoXhckCKSowMET1Ritd8Lrq9KeiYwWOJFLiBU4MGVKbyjxA7TAp9QdrXca2nNb4MT2kUqa1oRsVpAXU6aMj9XnQclI4kl3pXca2nNb4GwEULcVScCGYOSLJfr/bTICWeqdhu/WFw5FWj93WYEcb8qU0bF6HVgqAzYkP/bIKIHTWhmLVUem9MYSPUB1gdjdUKDUeqejMmFWbZkyOtQABzlfsvphDcLmBaxmpNlKvdMwbgROqaCGWHVkyoBY4ofAZ7PjO5wpIG3iB1mld7YKQ82PAqeSg/1WXZkyPtLJw3+RgT/GkosLhRgBAgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Question 19.Prove the following by using trigonometric identities:
2(sin6θ + cos6θ) — 3(sin4θ + cos4θ) + 1 = 0
Answer:2((sin2θ)3 + (cos2θ)3) — 3(sin4θ + cos4θ) + 1
(∵ a2 + b2 = (a + b)2–2ab)
⇒ 2[(sin2θ + cos2θ)(sin4θ –sin2θcos2θ + cos4θ)]–3[(sin2θ + cos2θ)–2sin2θcos2θ] + 1
= 2[(sin4θ –sin2θcos2θ + cos4θ)]–3[1–2sin2θcos2θ] + 1
= 2sin4θ–2sin2θcos2θ + 2cos4θ–3 + 6sin2θcos2θ + 1
= 2(sin4θ + 4sin2θcos2θ + cos4θ)–2
= 2(sin2θ + cos2θ)2–2
= 2–2
= 0
Question 20.If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p2 — 1) = 2p.
Answer:L.H.S
q(p2 — 1)
substituting p and q.
(sec
+ cosec
)((sin
+ cos
)2–1)
=![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATgAAAAqCAMAAAAQyLd8AAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///b6Il2hAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADu0lEQVRoQ+1Zb3+bIBCWLI3bmqZJWrst7k/r1kTj9/9+Ow4xKCiHipr+wguTF/Bw93Bw3EMQ3NqNgTkzkH9nbHPsYWFvgB5zjzn0xJ7V6fJodcx2q+4W9AboPvW4I2vEnRaHIMBPlc4Ku20WmgHG9WmK2eI72KZpWOMpj8jEmQGmcMXTnNkLY4vN8cRgq/LP35BtggA2Gsx33m1tEZfuYTiwyX/Xr9C7wGsA8OTEBLB5dHfMv0F4YXSl4f0hSIBC4bf4Kk2LuDR8CrLdMwyE3+jTGwwReA0AE3joaUoMqnRdEofsbSHWGDYbcbghocW8I+de4jUAePJiCtif7P4Pn7eIOCCO+37GhFqJuFhQCe2SMuReFr/4LfAMAFN453POf1/Y8q1OHPGMqxInBgm8D3/G8SV5D8HhasQFuAfPcHq1nnGGiJN4ZgCfETAuNj/ekKUacQne4+C0byVO7mWeEgT3JZ4ZYFznfM6G6XAPEYcXYPzw5MAPOagc+MmlNi2rJuwpyH8dYOAWsiryX+CZAXy6Mi529hIy9sAdhjscfng+BMKyPVs81m3RL8Bw7Vv+4CcbDOWlrcRrABjXuY8wW/57Xdv2vb1KHl57Y8weII8eSUoLRVIp+6RfjQUgBaONMDl+DguTR3ha5u9QmC1V4aCDJqPILulnjtUBo5W3UhdqWJgxo7RIzzGkkmynpmqSJlO1VJVdEp7MKRgO6oSCLxZmwiZvLViY8VK4qVEkFbWPyTMjhgNx6nhcmAlbRZ7CALFrMiTZhR8BJF1HJ46EP3XIVQTRWNFQmjUZouzCY5ii63SUdcTZbKrSxwlDlTh0067JEGWXas3TrOtoxFHxGx8PSsnD1x++NMklHxT51abJUGUXvVjUZZk+so7kd5wA02a5RFwpVFk0GarsohNnlmXqEeeCP4utGq/KNNWqyVBlF+WMkyeAUZZpJ07OhiZVxk8ccWVWFfeu7UVDadJkiLKLOCuF6C+JM8oydeKI+ELZnq5JMzG75zEQZ9VkaLILAlJ0nY6yjvZa1YVDawnY0qGsHDAFgQJl12RIsguPYJKu01HWGeIep7/MO9SIRS7tsl5tY4bwq9WmISoH/WWeUiNKs6jqiBO13ovwQRbGWka2d7hGPc5pYQglnL3OdIqb2XZ20uMIJaLb2/9saRnYMEIJZ68zB7bpGuBIJaKtzrwGR4e2kVbCubz9D23hTPGoJSL97X+mjg5tFqWEc3v7H9rCueIRSkR7nTlX57zaZS8RCXWmVwtv4FfCwH/u665RiIC8rgAAAABJRU5ErkJggg==)
also ![](data:image/png;base64,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)
Question 21.If tanθ + sin = a and tanθ — sinθ = b, then prove that a2 — b2 = ![](data:image/png;base64,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)
Answer:tan
+ sin
= a [1]
tan
— sin
= b [2]
adding and subtracting [1] and [2]
2tanθ = a + b and 2sinθ = a–b
L.H.S
a2 — b2 = (a + b)(a–b)
⇒ a2 — b2 = (2tanθ)( 2sinθ)
⇒ a2 — b2 = 4tanθsinθ
R.H.S
4√(ab)
⇒ ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
L.H.S. = R.H.S
Question 22.acosθ + bsinθ = p and asinθ— bcosθ = q, then prove that a2 + b2 = p2 + q2.
Answer:acosθ + bsinθ = p (i)
asinθ – bcosθ = q (ii)
Squaring and adding (i) and (ii)
∴ (acosθ + bsinθ)2 + (asinθ – bcosθ)2 = p2 + q2
∴ a2cos2θ + 2abcosθsinθ + b2sin2θ + a2sin2θ – 2absinθcosθ + b2cos2θ = p2 + q2
∴ a2(cos2θ + sin2θ) + b2 (sin2θ + cos2θ) = p2 + q2
∴a2 (1) + b2 (1) = p2 + q2
∴ a2 + b2 = p2 + q2
Question 23.secθ + tanθ = p, then obtain the values of secθ, tanθ and sinθ in terms of p.
Answer:secθ + tanθ [1]
sec2θ –tan2θ = 1
⇒ (secθ – tanθ)( (secθ + tanθ)) = 1
⇒ (secθ – tanθ) p = 1
⇒ (secθ – tanθ) =
[2]
Adding and subtracting [1] and [2]
and ![](data:image/png;base64,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)
⇒
and ![](data:image/png;base64,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)
⇒
and ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 24.Evaluate the following:
⋅ tan17 tan38 tan60 tan52 tan73 — 3(sin232 + sin258)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAXCAMAAABd273TAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAAAAAAAAA6AABmADqQAGa2OjqQOmaQOpDbZgAAZjoAZrb/kDoAkGY6kNv/tmYAttv/tv//25A62////7Zm/9u2//+2///brd/baAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZ0lEQVQoU9WSSRKAIAwEE/cFBRXk/z/VqeCNUF6ZQ05NMl0FUc2JG/PgdINouiuY9lAJ3+xEnlcVsHh8z70G4AIATMtfsDVFAJnZ/ASkQ/YEpZKTanHCsKQJgULHd3FYmEfNobo/9ACwvwRwCb8bHgAAAABJRU5ErkJggg==)
Question 25.Evaluate the following:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 26.If sinA + cosA = √2 sin(90—A), then obtain the value of cotA.
Answer:sinA + cosA =
sin(90—A)
dividing the complete equation by cosA
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 27.If cosecθ = √2, then find the value of ![](data:image/png;base64,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)
Answer:cosecθ = √2
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGoAAAAuCAMAAAD3NnzfAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgBmZjoAZjo6ZjqQZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kNv/tmYAtmY6ttvbttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bfPvyNwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB6ElEQVRYR+1Xa1PCMBDMFYEKglR8FYUiRJA+bP7/r/PuWtoUcWxa0xlnyIcMw+Syub29TSrEZZgzsL9fmQc1iUjm0Ns2CTSOiScb2REUnu0CZVygasCFwH9EoNONW6ilCwBXjy2ZsR6ulgDT0ApMBM/6vsofhok37AIqoiLzZH0EAyQvdiuZNgCNx5syKlkAONMwAiSQpncXpkIgf7gk9WYNtq+EaMQofxCqV0yBM4jd8UpIBM2gsrnBSD2Ufj6KIvDB45sCivFmmE+2UIMKvgfXOoR0jlVYw/iNQvKs8G9CT1l8NbIqT6/9qhyhhBK7EfTxVVCFalmrswTSAfYuplCFEqzA1NMU2IxATRZUJt73BEpyX7V+mehij90nkcwxK3YLnkgWVCx0CxOty7JY5+WSLNCSJ5QU9hRPxDNC4MPOua0lsWyRWnThLAyVPtjx5jPZRg273YC449IAq6z2c4C+bSLVC/ZmAKgxr7Vsf0k0axoikSzU6pBFX5zcsX+J+nmg3dZFhQJbBCZLdv/07vhx1f46/YmHjwNnUUhd+SYOY8gu+5jMpVDj2jHcXl+u/N62cKVgaNUzUHIsdRI6XT2RPQrxebLLdo+vUYcqsAeFTTvKSpXfnhahUuteVGpDdufqLfTbNvQLdaQo0SadSpkAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 28.If
then evaluate ![](data:image/png;base64,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)
Answer:1 + tan2θ = sec2θ
⇒ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGsAAAArCAMAAABIOYZSAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bf9mRtgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB1klEQVRYR+1Xf1PCMAxtUWDyQxjCJjIVh4yWfv/vZ5Jua7lDxijZH565Y+tB25e8vKZBiH8LYmC3CVretHi/oP1VJMF6nFg6lg9fFmvV5FXg72qyzbvCAlf/PNYokn1OaTgOTboUOmHVYZ0vEpmKZoFiu7y81AZNOs6HXWBlGBF7XDZHGbx0Ys8aj5k1lqbHFyH0AgaTLQ/M/XY1aymnxf32u7CTSYaFZhZVBX/AhNOD37IB8Bd6TaiRLxUVw/UG9w6+x/iDTuGLaQEU0skMrAI+MSqCGjZfQQBYy+BsmGRQmFf4EJZ9trbjHG9oay4LxBSeQ9wTGaNI1LiopntY2Zn1zW7kyBpaxZF90/Ndjj5pfF1cLgA3OnHgFyzL2PeT7COXAfk6y+GZuABtHwEOsXuEVNZ2C4eeNqrcoyJsviBVFien8xVYSU80n8ulMB8bcZAz0CFAkCBjiAsSBnUjUPKn0tlBi/CGWYJqjdVPpzCYoDih7+s9N+us4xll35tbJZaniMWHuu/NUEehVe2ii67vpWl+78ERmrd/cAFt8s/DopPCaQ7rxlrdwjmHdZDcf4scVnVttHC05dQai7sR9XWeBZbPK2KsxMcueNf3dtV1XRE9z5QfrgEmGQqcHLAAAAAASUVORK5CYII=)
and ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 29.If
, 0 < θ < 90, find the value of sinθ and tanθ.
Answer:sin2θ = 1 – cos2θ
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 30.If θ is the measure of an acute angle such that bsinθ = acosθ, then
is =
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
dividing both numerator and denominator by cosθ.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGcAAAArCAMAAABSBWbcAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOma2OpC2OpDbZgAAZjoAZjo6ZmY6ZpDbZrb/kDoAkGY6kLyQkNv/tmYAtmY6tmZmtpA6ttv/tv/btv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///bOPDNJAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABzElEQVRYR+2Wa3fCIAyGoU67OW13qc66OdlFe/n/P3AJCLiNFMHqB4/50GPanjzyJtCXsWtYBdo555MNoYjgg3U/YrWz0abOR1QxEcipbhfuUtsEHsiLM0I5ZKVyCJpVaXEcp/28A/UZW3KIwVqnW86LjxQfgGxAaPKM5LzPefLs6RGK35ZGGZtW6XjBBC92HEVzhUjuV/JFf0hRrG4y3V0yWAmulHOaA51r6DnZw5McVEuV0OspFRRCD4aaA9nErmi/HlPohV6PSSUbOf7+HMLBhiiNpG42NRz1V5u8e95865HVLcemliPklJC7UT6mx3GnZZVmrH5Sjc/Y64tJt6gl3sMGwXlAjTWMGkzsG7mNdc9g30y+cyy65EmB20imFTYNRw3q1w88mZI9FlM4/m6o08I/7Nc3LlmBarw6z/LoD1YffH3o7h+AfdTtqiFgO8owR29vP35hDecECzq/bqedAyPR2eY6sCmdXlTXOt6TeryoAR3oScO8aDv7+w0/1JOGedFgTqQXdXG6PGmsF3Vw/J40wou6OF5PGuBFcbr++1G8qz0psWkivSg1B+XQzYn1oqGcWC9K9ofwkrFe1MHp9qSRXtTBuVBP+gO6kz4Gj9GwqAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 31.Which of the following is correct for some 0 such that 0 ≤ θ < 90?
A.
>1
B.
= 1
C. sec θ = 0
D.
<1
Answer:for θ = 0, secθ = 1
⇒ option B is correct
![](data:image/png;base64,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)
Question 32.If
, then
is ….
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. 3
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE4AAAAuCAMAAACxgN91AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpDbZgAAZgBmZjoAZjo6ZjqQZmY6ZpDbZrbbZrb/kDoAkDo6kDpmkGY6kLyQkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29u229v/2/+22////7Zm/9uQ/9u2/9vb//+2///bRGXQrQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABg0lEQVRIS+2WaVPCMBCGs3gQUaBajxaRVgwg0sTm//85d1N7MBZJ03zxyIdOpjN5+u67R8PYL1+bcOEvQnUDJy/ecHK8FB5xqOsf556cv+XdwF9X6DkHgNN7d+t/8kl54c9J9CHzmBjGUkwMTiy9HgFMEA4QrTjtHJdRp+PhTie0k/xywQREjrRGsJIj5PMxbeASCsAsC19q72pcHjRxrUKrT+xtylToTYjlXaqzwB0yo/KujBOZ+7juwRpmp2APqZN8yp6f8KHuKBVZEfFR72qcaLiHb1MYRFR+k7cAIkkO5gGANU/PvHZVfrtzrdG2c9mwfLvyITMxXlsW+dE49CPeUExNeFnyCq3zhxNUAj5w71sKLyX/++PUHCil+TVd7nCqcTjrk9jXbUL3uqJM9OyBqdhifH2TLtOJok5pp878ytUx/hMaHZYHVUE71QxOC1MmjCVFeq37vPVzOj5fFwT6yygU66SqOiRgVFinQmyy8bIfDYd2X0H7AkQ/83tG0/n4Bx3+H23vYWNyAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALYAAAAtCAMAAAA0j2G2AAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZrbbZrb/kDoAkGY6kLyQkLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bgaTg8QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC6UlEQVRoQ+2Z2XqbMBCFJTc1XVLTtDFtTNpSpw2L3v/9OhoJSWhBCPkiTtEF5ouZw/FoNKA/hGzjCjPw547S2zzfF5BINTB8fiDn3Sk1zLz+AhKrbt+9E7bZN0j8c5ZEjkbqjZuDcF3tn/tynxqN10uJLI1lNx5zTM6ytFteK3jwDxXgfj1KiPAZjWXW5q4axZtxQdZvoUC64hgKCrtREiSmkWy7g46xA0v88+NPQh4pjDe/SAuuW24VagSOQykqxh0ygJ3fY+tpKT3+LrAJKYmoRrrr4p705RGyCZ8V2JVTOZTcvrYtzHsHZpsvAFbzs674cCINhGoJaXtGI9U3zh6MmrvCSrDnXNyd0nnbKILx8jCZnKhGou1x7sUnHl3b3LDKVC1+BIxxkeoAbduqqQEb0eWyPbWNurbtWG3LAPb0pcCqQu+W7ajG5bMtusAA9R+rbaNI7BUc00i0Pc4bq1SXc/pZgz2Xr9aZJYlRwSIhMY1E27Dk7wn7cYK+dYBOIrwfCPxBDyhMeEqG+p8M+A5R/Vdc0qJSptfHNFJtE+ixNw8QdYbKFC8ej9jHjdHf0d2nGWEMgPZ9+7ekx45XOO8cU98xjWTfW8CWgS0DWwZeTQbcd53r/Wnqte2FnlxtZl9TkVztJPynxg2gl0GpePJMNBiTYk8AFG7WMz0D6OVQKnBtosGoVA3bgb60tyisCm61VFFp6CTP/ITJkZqBVWT8LgqrEChwejEZS2zrTZoEen7C5EjNobMRDS6DVbhfMofHdohSwf5H0LXALtyWCsMqpbQUVtXxIkE65aFUMFWSCQrbDvPwZ9sDq7TSQljl4kc320FKpYBegDDNFIkFqzQaXASrWDXZlXqfpkFKpYGehzB5pUKwykCDS2CVD2U5KVpPqQLZzoZV9d79P0bEtpcJBihVuLazYFXD0U1rQRrnXuspld92LqzCBs/qmO31lMp93HC6lQur5KqJ2l5NqdymtMGql/r6+g9gd18/Aa3ITwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 33.If
, then the value of
is ____
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:⇒ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∵ tan2θ =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAEVQTFRFAAAAAAAAAGa2OgAAOmaQOma2OpDbZgAAZjoAZrbbZrb/kDoAkDo6kDqQkNv/tmYA27Zm2////7Zm/9uQ/9u2//+2///bh/RrqQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAaUlEQVQoU6WQSxKAIAxDi8hHUREF7n9ULWVQBwcXZhXCa/gAtOQ1Y71LRDQSvJLJBzUC2C7nwnlco5AXxAc9nDPE7PzysPLS2Tz7Y5OR/lRUs9FgZX7X7CBOS2E2UewjJjr90J1+j+vbHtBKA1mNxNW5AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Question 34.If
, then the value of
is ____
A. 7
B. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAA0CAYAAACKJRg7AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADkSURBVFjDY2hsbGSgJmYYNXCIGZgX7eHlbiw3i4HB+G5sfb0kRQbmuMkWy8oy3GRgYPhPFQNhOMqYYeGogaMGDisDOzrSeDxMGNaCDPTIqzekyMCGaGMfSLZDwsZRC0fLw1EDh5yBGAmZAB4Nw1EDaWlgfZ6HoSwDwxvs6U/2Db7SG2dDCWeCJlBqY69HjI27I/PqVbFVWMbRDT4kGVifF6mKzbD6+lhJIwaj3YRqP6IDG1xpUbOSIsa7RBtIrHeJNpBY7xJtILHeJcpAUrxLlIGkeJfo1hex3iVoIKneHS0PqYMBRwUDJFMu1C4AAAAASUVORK5CYII=)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:dividing numerator and denominator by sinθ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAqCAMAAADlP6mgAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOmaQOpDbZgAAZjoAZjo6ZpDbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bAaqEMAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAiElEQVQ4T2NgoCEQ4xHEZ7oUNyOzCB4FkhzCongVAPWOFAVM+EJSRoCdkZGRhZeGUTnIjBYC+hcC8AYMmqvhmhiJ8A55VhBhMMVKRCH+YBXHZZIQH1BGkh1E4gGEso40FycBAwiEvQw/G34DJBgJOFEItx/BJkuyE3CiEIHyg6AfJUhJX3h9CwAI1QUSGUaZqgAAAABJRU5ErkJggg==)
Question 35.If cosecA =
and A + B = 90, then secB is ………..
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAA0CAYAAACKJRg7AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFRSURBVFjD7ZdBasMwEEXnADlAAlJ3OUA85AquCFmXxsK7klUYKDmA7J2h5ALd5RzNtpveoIvcIHdoqxRDaGJnxlFDC1rMyvCYb42kJyjLEkIWROA/AToyIw2wA4CP49I7Q24kAi5SvTwN+yrM1qIOq2reM4hPM3LDn98yhDXaYioCOpoNT8GcywcJJC+5c4Mgi1JYnJ6LKwJy4rKB3LhsIDcuG8iNywJK4rKAkrgsoCTuWaA0bjwPI/BPAhvvkYaK/zACrwH00pRoeKslCVN66Az0sLGxj/Oq6nnfsUY7Dxa7TS1MeZrfHUmUgo3YvpqqBup0sQwC3EsoZquLV9l3RnZ8r0G/G0uTi69RBNgenjBBIhdE/Qz16huK27Z7WjS03iI6SXur5wQHKrPxw97pnXKDt8+1wJM1E6XUa1t3rTtlvysOHjt+BomKfjwPfx/4CYRaAUPHp+vdAAAAAElFTkSuQmCC)
Answer:![](data:image/png;base64,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)
∵ A = 90 – B
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJcAAAAqCAMAAACJFTngAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZmZmZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLaQkLbbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bznCg2wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACjklEQVRYR+1W6VLbMBCWnDqYlgAxtA2GNDHlMk1s1Xr/d+sekuOANTgePF1m2B8+pJX07beXlPqUD85AEa0lWlAlInHV6XeRuPJ5KRFXOdtKxFVfrJVAXDabK8RVXMnKyFI7EYaLWBLoR9G4hNZ7iLEBBczez54G5Epx+jhgVf8lNjvf9tduaVbfxkwxqi9KbS61jokAu9R6FqLCZpT10RmaUh2PeEsoJ+jEIlopm+OnzaZbk9Fgl9TpEejcaXjCqukwpnu4x2Z4Qp0iafSkUlPqkIu8PiE6mLAKvBLB1vgmn5hrRz6MRD8Rr1OpEkTAT6RKEWnESqd4XDTvYqCH+axSJQtl0it+o0/wRHsDp+LIBkF4Fa7EzBdpITb4D3mIcP29idnPeQh+N9Tc7UrLkAs6t5oBGzBDW3sV1yHyyRqjxuNidF3CgT91eeF3YZC+E4aKFVu/FzW3+uShGYHdvIoqOMIxCeNf4D1G5HG9PoqMsrkrk3u4ug1pwGrnlZ13kLU/XzVwX6ekNnlqcLU7apXMHaK344v57IGrhbaDL5jdJEcN4t1HGxd1MRf3zPhr4bi3rpDsxdebfuS13hsYXxhaaJyf4RhD4Ux0bsexAktEzzrRsB4w4uVwoRfK/obrq55DPro0NJdwbIkzz1AovEoDEPikIo6pGA57riA2h43RqEML/nOi4xVGVaI1nmau4eMUj4WZ6If7IBWu98DbCf6BGKh5pNolrg99WdDkiPX+4NrYQtuPLgspp+N1T1fv1Ma+T+TQek0m7/5FOUvZJFCo+ogTsxz5bjvIYmwywQQftOO7LdokIv2I9TzU2N7N9kEb4ZVAlvCdNtxw/xdaiyWVmrQwMUvXpIXhEg/nH+yORzrsROJWAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Question 36.If θ is the measure of an acute angle and √3 sinθ = cosθ, then θ is ____
A. 30
B. 45
C. 60
D. 90
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 37.If
then the value of (sinA + cosA) secA is …..
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABsAAAA0CAYAAAB7LkO2AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFpSURBVFjD7dg9bsIwFAfwdwAugNTHxgHIK2vHYFUcgCRiQ0zICwcw2SJVvQAbIwMnaE/QG3ToDXKHtglCrdI8N35xPwYPnoj5yX/bsV8gz3P4rQYBC1hvzGg1QYASAF6/NiyVNhNv2CbGbTv03ig9eBtZUawHiugu0Wbc/C0lOFC2m3vDjE7GbZAxy2EE0cPSmOGPL5BdRnOXCHthrhGKMUmEYkwSoRiTRCjCpBGKMGmEIkwaoTPWJ8JwxAQsYH+AsfcMYQtzFrCA8ZjO1O2MRnsAerEdlDqbLj4KDiyR0vt1UQw6Y1URgQjP5z/gsfou0raZr9QjB1rvGhxWFxuIx2mmF5/TuIySu6OIsKpeU4m54UbrFbPXcXwfr1jVB+PN1nnpu2LnUjh6sj3vBbssmO9qay+YjmnV5ZbcG6v3pWWevGH1iGK9akY6G9G+rR8/B9dwqjBuHiSfK7q9hhqdrZBkU4fz7N9ib1udEZmNAunuAAAAAElFTkSuQmCC)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:(sinA + cosA) secA
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 38.If
then the value of
is ….
A. ![](data:image/png;base64,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)
B. 3
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAA0CAYAAACKJRg7AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAF5SURBVFjD7Zc9bsIwFMffAThAkTBbDxDcrB2DhTgAicVUxISeVHEAky1LL8DG2BN04QTcgRtwhxQnNW1p/BWiSkge3hLJv7wv5/8P5HkOXQYE4B0BBaaPIwIHAChlDOl4uyyKXiugQBYRgFPMcabgbAB7GLC9DfrnQVEse9Vhmu1+vUTM+xTgSJLV2guosms6mFHYAdDjXIh+J8BVQtYA5MRQRO7Ar9Ka+lVn6An8PgglodmbgiKmUT11z5JVYEIXsvRqbcjowDh70WXearHr/kFJ+WZ6800x9dUbeLkx57JNvbMCN4gPnMWvVR8dMrPfFDll8vTBOE7C9zAA7xqoBMk1Qg8D8D+AyOPZRfnOevxTVv0lgNNp40K3kdFKBgh5V+6rzpZNVLYmKdXaOZaKZ13W3kCz2Le0IjrP4+0P9YIvbZ5d7J1gakgmG+cFlE7MZpKcgXIQtr653xSZWYKL6/LHQ7rV9dLqBxvj6g/BCjTCulzs8IFtjE8p/fYU687GPAAAAABJRU5ErkJggg==)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAvCAMAAACmCbqiAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGZmkLbbkNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bma+IXwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC3klEQVRoQ+1Z63qbMAzFbFnp2mxJs0tLl6V0a92kweD3f7lJ8gWnmMWFsDjfqh/9SArmcHQkHZwkeYv/nIHnBWOfToWD+mqVrNO7U4ELOMX5KaHls3ip5S/zvo5YtiJ7gZZHCVb+fICE1/OviFZYiCUclTfRSeGe1FnMSuJWfUL0DCI6tPwMwZWXW4VW5tEhdPKrulR9dZcotEn57jG69FtABVIrc8g/oOXAKx1HGoraEjVqdBoxufzD1vColQD1FSu5LjKDNimaJ4hLERYhwLKzzP3y9XBH9G68aQCgXT3MRDZACiN6N5n7ki5zasH9w3g3uQSLbOui/3r6ynruxTVUuNq7wVNvK/8dWsg3X6jZ/zU6cl4MGxDGu5H+g4qgWrD2LVtTtfQbgZZ33PfUO/+33o1SJLL9k1xMH5wCMosNQysu0KvpqG6hKFGTYsHS7/glHLAUkFnvpuQf2L1D0HaQ2EG5k1QsT/kDuBPZdbJB+vCgmt843k2hDSzZQ6FV1lGF9evEmLjc0iQhQO3KNBe6hVy0l1LJOhRaV5McU01xzy5+2zwDUk/CVcPpx633qYwSGvYISocScNzZkll/ZJNHnXcoaA/ao+jWqwR8pk121tSQr5hIHDVouYl/qASnylCyKFWbZ1/CKXOB/jNEt6/qt24How6wAF2W7DqRT9DCOB78AnhNgHBhlkE5BoSnOw3rt+49q9uMsSlagKeMpd/0wWS1iwtGVPo5AKpcwmrsPbXtJlpox5m8Afh6nWJdjYQSZxOTx6GupheW/RdZx1ikq6TKdfMPbJH7lz/0GaYU6c2Xa48zyI0fGqG73o6fM0YxyOSNiaprbaepV8up9liRyhazr191cGJRRwr2eMcg19n+3mTKvwdOn2OgTai+VJRsBn9j3llyf1tQvSBmakG5QK7ik1xD3DuitL8scTBUtLlgdpuPIsuAm+JOfoW+gl4Ko/yxIeAp3k4Zm4E/SVBHbZ1RWoQAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
0![](data:image/png;base64,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)
sinA = cosA.tanA
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAAqCAMAAAB7ha4AAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpCcOpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZpCQZrbbZrb/kDoAkDo6kGYAkNv/tmYAtmY6ttvbttv/tv/btv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///butK5fwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABjUlEQVRYR+2W63KDIBCFIU1DTdLYu1jtjbSi8P7v113UaKwN4sx2+iPMhMkkwFm+PbvK2Hn8TkAtMlI8paAVMPEDrUC+06QCel2QCpibjFEKWLljKKASIh9p3gwqARc3JaK/EaCuZMgDaSEQ2Sf0WM2HPrPysgg95cT6nwKaGmz+KKBI54wyeuu2VSlYYFuAERKG017wrfvX3L7PZtQrPARtnwF2KSAHpYgypupsqBVTFx8hFzBx2zZ6tjUxYCjXBwGngmSsTGrRbuSH/X7Xq0W79YVHr3hGcwP42Wk6UStX3ht0QR8t7QTY5xVfAomhgHI7gxiNIkLZLwGBDgTwMQJd8qg4piPqJRlJsLxLcotIuyIzsZ/RCMS+TUvxxKo7OMYF6yZMckPfxEGMxhJWpYLzDZoIasBNSPIaPnAF/DpRYU/35oVRkbbrQbF4fR284CzgRQYNUvAlnYmYTaGSpL/9eSM9uaDuwIRjZk+ZFFHumjvhDXLAX8mJHWVSyINF1T10ik3vKT/nkP+x5xuYAR5SclGRFAAAAABJRU5ErkJggg==)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFYAAABWCAMAAABiiJHFAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZjqQZmZmZpDbZra2ZrbbZrb/kDoAkDo6kDpmkGY6kNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///bcFOa3wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACO0lEQVRYR+2Y61aDMBCEg1alrUVbr1RFK14aIO//eiYtJNhiMpuGY3uUv4SPZbOb2YGx/+v3MlDEEfkCouVnwCL6ksPC5hP6FwJP7AE2P3oCAl0tye/RlayIcWwGY6vkug9sNuE49hlNFx8tcayYg9jq4omFx4pU1rfEgjuMRsub8wjb4eryFS4wQhL2AYt3WXW11En4mNmrQuYXLdxi1GDLaXRMyLN9RzS2GC/ygNhz89qA2LbmHCw20yJf770Rfah72poTMAlE7Bt43sHY9fSDtkNbc6y9WcTY2bVOvcaKRxXO4OanHfHEujaYhoWljIRFxYHJIWEYRydgIeBYMb9jZQpWgnggnYVFjA2CJM1hrEqwIRvHZirO8NFmMq1lCh70RnNcdVvOZLeMF65l6/s4doO3dXJ+u++NdSikkrLwvszbPtmT4I21J4FjXdOCQAMT2T5htoSKBW0JFQvaEtw+rdKL2hIaFrYlJCxuS2ApW6UAtSW45ujKRbxZX1ialKmQkS4zmuOyI00SIFvSYDvtiEgpU1f77Kmx3XbEH2vEoWNk/jNY7co2ktCpKbAtMZoTNLdeWLctMVIGRYvaEiMOHT25XWDoRF5ju+3IDlhae6LRksRBjbeOMMTLWP31IGPttuR9OI3Ui4lYty2pklMZLknK6qPPPujnMlwPzXHZEhUuFYvYEhku0ZUxxJbIcD8pPxhlaiFbkke3RCxkc6pkoH/ZQQ+Ai3ikiiz4JdJesIz3gw3+/TsCvwCzMTnPyhh3yAAAAABJRU5ErkJggg==)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAABBCAMAAABRqYxcAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZrbbZrb/kDoAkDo6kNv/tmYAtpBmttv/tv//25A625Bm27Zm2//b2////7Zm/9uQ/9u2//+2///b3dILCwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABpUlEQVRYR+2W7VrDIAyFi3Oyqas6tbMqzkLh/i9RaCktkH7Q+Dz6Y/3Z9SVZyMlJll2e36oAJ+nPMDa/wWXCDn/Ll8j45SsufySvTm+o+FhePuDiy8fPYf5fqac5XlDTx1dTPAN+FfuqjS/o3EUKiuJl/gzwfLs0fnngEN/Jby5/vq9m+B0l16PlM/cM8U6+6vSS1UWXYemGSvtGFVpmmmdhkX35CzqiRjelQt6Xr8ynphGUv5Nfc9Bo/KZFpnn9R+ti43VzoEyg/3r51UddstuPCTHrGkQN8M/kmzyKAvkn806+yaRVfSf/lbyT71oe6Z6N+yYbeJ/sxX37WiR7jx5rrfsu8J4ujD8DHA95jyrit4EHWfeFZz/Ahx5k5beYDz0okY88yMpXUMh7ovxjD7K87z2m1KH/jHjQUL7R7A/jAx40XH4j74HuL/AQy8Pes4C38i1B7wF523/q/c6UxPKw90C89aDz7omY5ly/PAtqFh/E8stMAghe5tsK5Z46ARSvE/hGLe+M3PvLc6IHyZxMriuzx3HSLa+zn4IfqALHZ2vd9wcHbiU0xuqSLAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 39.In ΔABC, if m∠ABC = 90, m∠ACB = 45 and AC = 6, then area of ΔABC is …..
A. 18
B. 36
C. 9
D. ![](data:image/png;base64,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)
Answer:![](data:image/jpeg;base64,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)
given ∠B = 90, ∠C = 45
by angle sum property
⇒ ∠A = 45
⇒ AB = BC
also AB2 + BC2 = AC2 (by (Pythagoras theorem)
⇒ AB√2 = AC
∵ AC = 6
⇒ AB =![](data:image/png;base64,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)
⇒ AB = BC = 3√2
Area of triangle =![](data:image/png;base64,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)
⇒ Area =![](data:image/png;base64,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)
= 9
Question 40.If cos245 – cos230 = x ⋅ cos45 ⋅ sin45, then x is …….
A. 2
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:cos245 – cos230 = x ⋅ cos45 ⋅ sin45
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 41.If A and B are complementary angles, then sinA ⋅ secB is ____
A. 1
B. 0
C. —1
D. 2
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAXCAMAAABd273TAAAAAXNSR0IArs4c6QAAADxQTFRFAAAAAAAAAAA6AABmADo6OgAAOmaQZjoAZjo6ZpDbkGY6kNv/tmYAttv/tv//25A627aQ/9uQ/9u2///bgUiD0wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAARUlEQVQoU2NgGNpAgIMbnweE2BmZ+fAoEGTj5cerAKiXaAU8jDDAhOoook3A6VKYCbS0As3RqG4R5mIFeo6Fc2inFrjrARYuAgUBsHLEAAAAAElFTkSuQmCC)
Question 42.The value of tan20 tan25 tan45 tan65 tan70 is ………….
A. —1
B. 1
C. 0
D. √3
Answer:tan20.tan25.tan45.tan65.tan70
= cot(90–20).cot(90–25).1.tan65tan70
= cot70.cot65.tan65.tan70
= 1
Question 43.If 7θ and 2θ are measure of acute angles such that sin7θ = cos2θ then 2sin3θ — √3 tan3θ is ……….
A. 1
B. 0
C. —1
D. 1 — √3
Answer:sin7θ = cos2θ
∴ sin7θ = sin(90–2θ )
⇒ 9θ = 90
⇒ θ = 10
2sin3θ –√3tan3θ
put θ = 10
2sin3×20 – √3×tan (10× 3)
= 0
Question 44.If A + B = 90, then
is ….
A. cot2B
B. tan2A
C. cot2A
D. —cot2A
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARkAAAAtCAMAAACtUu8qAAAAAXNSR0IArs4c6QAAAK5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpCcOpDbZgAAZgA6ZgBmZjoAZjo6ZmY6ZpCQZpC2ZpDbZra2ZrbbZrb/kDoAkDo6kGYAkLyQkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttu2ttvbttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2/9vb//+2///bPm2KCgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAER0lEQVRoQ+1aa1faQBDd0KpptZZY+wR8FLTPiK2EJv//j3Vm9kk2k2Qj8eQckw+AMDtz587sbOAqxHiNDIwMBDLw5zyKTgPXPAvz/OxarCfLZ5FreJLZq70wUywOH2qCF4sIr8lpnVED+IYQ4nExNtG8FD+dmjeaQlcgN7xuIq73pEmeHAlRfIvg0b2KRRmPF8QLwdaSidGuWzxm1s6Y4bNjnW80H6uPsaV4x1yaFAvkJE9KndWCGS+EeaOMionRjpmyVerOXz47vSyDiT2BKuPzyU8hbnCLvPgNKb/7pbYTY6JRN/ZMYwgVs1i/ptNjE0Xzu5jOESZGPTPbC9rk6Mb1tUHXup9tdpyvLJ6JbTIX9LxARnT90iOR4t/yowoTQv3v8gBtnMvrmRYhKCb4eyhW+CqLj5cixSnBxKhlBidIcQmdnMVYce0rT7DmmhmTHetqpfbCCitPrhQzmCD9LTgTOR4Poc/qmWkRwu4mlQ2lBHuZiVHLTJ7AwuzEMKN9lcsns+Mu8kLTAp/pUaFE11Qx1oQ+lUXW14qOKzqyzLstQpg+VcUhzLSuIkYtK/ThTXT8A58tyxqDWWuya8kM4VDMpJQgbKfdzByTlhOYXQ/7Rc80NdLv38fU7yVm/CnfwA7MK9zjNczY0ME9k59h0fHcq+8ZqCoNI3uV5wy73oaQ1cA5U85Gsu/FaOyb+xjW8cw4oTlXMjLGxnnjzJkNzR+8nWBNWvZMixCSGXooZdPl1MadQsORZ8Zmx3OcRjNR3C6hO6ZwNkl6puL2q2QsT6AhGJOlvAtbwcLanuHWuyEo5hU8bD/QISC3FDhmYtT2DJ2F5zgX0I315ba1zY53BTcOB9fw8Rq2uLzRv4kmX+CEA5bwoANqKk3m6s795azk27/TaxECYs7xVur0bxLNM5w2GHvKxahlZnsB699gy4Ab68uuQdc2u8adORqMDIwMjAyMDIwMjAwMhoGqb3uDATc8IOa78dBfSOraVDc0k+EVZTiI2vA9HLQjkpGBkYFhM/A00ravADbJfo9THPfAuZW2e4XiM9Mo+z1KcdwDM+hCyaFPC6VR9uvy2+WeCDFulLTdBxRGAXRETWVBuufkMxVKCqGdFMdwasqaqDCA4KdLpeD2AIVRAAG/lv0ci5m4lzqhVDm7KI4diClrqgYQYNTSdg9QWAXQiJrGAn+/JwRapeyiOIYz42miBpCw0nYfUDgF0Mp+ykIKTYDTKISdFMdQair0LQ3ZkbZ7gcIogI7sJy0kDhAfdpkJVhwDqanQRIWC7HjqYwKD+yoFcFf2QwtDSImZcMUxiJuKntGQPWb2CoVTAK3spy20OGlUTu5/kYISbzT2NVEDud+eYRRAhQdFTccC9M87OLa1ytlFcWxkwjPwNFUDyDHtAUq1AvjWkf2+K41QoLg5+YS6p1QpOymO4cx4mqqGvNMyNAM9gbVDtHHJ82PgP814xv9VfkmFAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 45.For ΔABC, sin
= ……..
A. sin![](data:image/png;base64,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)
B. sinA
C. cos![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAA0CAYAAABhEqM4AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFmSURBVFjD7ZcxcsIwEEX3AFyAGW3uADu0KR1NhgOAPXQMFaOGAwh3brhAOkruQJuGO3AD34FkZZx4HLBlSxQZVGz7rP36q/2GNE3hUQUBHuCmtglNKdlOvcO1XgzHgCep9Mg7PCbYA2DuHb6OcCOE+ASg80LroTe4VnKEKA+JRO0VnmWrwdsLfTDQyCLkcZVlAy9wFdGydIdXuJGD4l3ZgRRwBIr3zj7/gQFcqoXReuMMjwl3Vcuxxwng7Aw3Uxip5R+JAPK26WyEq2QyY9vVL40vlmWpf9QaXkzgVd+rK25q3+KY8J4H+NPB649TlwqaB/h/hpuN9L3WCj9jzimgLVJY79CbQ2ORWRrhZq0hHiaJmv12Id/LLpwWNG95Odev97pxgjelXJsQ2gvOycAmEPWL0DA+2Zy6c4TmC277m3CO0F7hfIm2OnebUD5xLRdW/zJ6wwvb3VltLuG/EfyoIQrvuXN9AZG04rMyqIaoAAAAAElFTkSuQmCC)
D. cosA
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 46.
= ………..
A. 1
B. 2
C. 3
D. 0
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAXCAMAAABd273TAAAAAXNSR0IArs4c6QAAADxQTFRFAAAAAAAAAAA6AABmADo6OgAAOmaQZjoAZjo6ZpDbkGY6kNv/tmYAttv/tv//25A627aQ/9uQ/9u2///bgUiD0wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAARUlEQVQoU2NgGNpAgIMbnweE2BmZ+fAoEGTj5cerAKiXaAU8jDDAhOoook3A6VKYCbS0As3RqG4R5mIFeo6Fc2inFrjrARYuAgUBsHLEAAAAAElFTkSuQmCC)
Question 47.If 7cos2θ + 3sin2θ = 4, then cotθ is ___
A. 7
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAA3CAYAAACCayPpAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFySURBVFjD7dg9bsIwFADgdwAOAFIeWw9AXa4QLMRctbGyVUzoSRUHcLJl4QJsHKA3YO3SGzBwg9yhlRuBUIHGQGxH6A1ekiFf8uz3E8jzHEKv4ABGMKJRBCk5Hon+EkBsU6173hGzGOeIsAGA72CI3UoErBjBCEYwotWIoph25BN8GIQkPfCOyJSYVCn7YIlkxaWcEYxghBfEUQJytO43HNd+gftEtCIcQRG3AhjhDKFJDh4RvqrTgKWI6c054i9gKNX7tCg6pv9UErW5L1Q28YIwD03j9PmoEY5gbdN7OtsPOwTGs3kwhAkPimThdE+cA5gvQGr4goAbqWjsHaF12hMA28N6cVM46grPf6HIiLqJwIXtP63atzzbiFgkqWpWxbJuTLQu09dkyWpcbABxCnIRIpJrk8CcI36PI0DZF6PlK+kHc838bI2i6NNmYr+4ezqF2GfH/WbG0uQIoqzb6BG1adO8VNFWIJrqH3gMbDXiB1Xab7ex+kGYAAAAAElFTkSuQmCC)
Answer:7cos2
+ 3sin2
= 4
⇒ 7(1–sin2θ) + 3sin2
= 4
⇒ 7–4sin2θ = 4
⇒ sinθ =![](data:image/png;base64,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)
⇒ θ = 60
⇒ cot60 =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAhCAMAAADu4HshAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6ADo6ADqQAGaQAGa2OgAAOgA6OjoAOjqQOmZmOpDbZgA6ZjoAZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJCQkLbbkNv/tmYAtmY6tmZmtpBmtv//25A625Bm27Zm27aQ27a22/+22////7Zm/9uQ/9u2//+2///bt7j0xAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAApElEQVQoU61RyRaCMBBLEcUNF1yLYusClfn/D7SLSJ/1BObSeZlmkmmBDqDTKFDJ2SIkAfF/kvJh+W0vmMa8wyp9JcbXR995nt7uZODPPF9Ch3r5w7VaAXLCIn20kBy0PeIa8ZajvXtNNXYkFVqi3Mjnbu0u3ou4RJWZMmeDNwnaZGgC3RLb1RDxw6jrlEN9yDo92CQy8SMJP0gTSU2D7/V30PUL2HYIyospqScAAAAASUVORK5CYII=)
Question 48.If tan5θ ⋅ tan4θ = 1, θ is ____
A. 7
B. 3
C. 10
D. 9
Answer:tan5θ.cot(90–4θ) = 1
⇒ cot(90–4θ) = tan5θ.
⇒ 5θ = 90– 4θ
⇒ θ = 10
Question 49.If A and B are measures of acute angles and tanA =
and sinB =
, then cos(A + B) is ………………..
A. 0
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:tanA =![](data:image/png;base64,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)
also tan30 =![](data:image/png;base64,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)
⇒ A = 30
Sin B =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQoU2NgwAskBVhg8kJs3HA2A4MgaWxJfmZRqEGCjEDAgd9aImRBpgABESpJUCLCzcjIDlEvwcXLIMzEB9cszopgCyKcLwxVDlQniGCKAZliPBBzOEHOhLCxAACkwwJsDMBvegAAAABJRU5ErkJggg==)
⇒ B = 30
⇒ A + B = 30 + 30 = 60
⇒ cos(A + B) = Cos(60)![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAAgCAMAAABAUVr7AAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcUlEQVQ4T82S2w6AIAxDh3dFhYki/P+PCsbnDUww9vk07QoAxeV1w2RsneQQAPwP4lV90CehCBqKT5sfoGKxW9Wa435M6Za3QekJH5C7FKInc9y0gOGnPFt2bWS/i6GrhJ7IEjYQdqZOcmN8axLJWP4Ca94CvCm+VXgAAAAASUVORK5CYII=)
Prove the following by using trigonometric identities:
Answer:
1 + cot2θ = cosec2θ
= 1
Question 2.
Prove the following by using trigonometric identities:
2sin2θ + 4sec2θ + 5cot2θ + 2cos2θ — 4tan2θ — 5cosec2θ = 1
Answer:
L.H.S
2sin2θ + 4sec2θ + 5cot2θ + 2cos2θ — 4tan2θ — 5cosec2θ
= (2sin2θ + 2cos2θ) + ( 4sec2θ – 4tan2) + (5cot2θ — 5cosec2θ)
= 2 + 4 – 5
= 1
Question 3.
Prove the following by using trigonometric identities:
Answer:
L.H.S
Question 4.
Prove the following by using trigonometric identities:
Answer:
L.H.S
Question 5.
Prove the following by using trigonometric identities:
Answer:
L.H.S
Question 6.
Prove the following by using trigonometric identities:
Answer:
L.H.S
Question 7.
Prove the following by using trigonometric identities:
Answer:
L.H.S
Question 8.
Prove the following by using trigonometric identities:
(sinθ + cosecθ)2 + (cosθ + secθ)2 = 7 + tan2θ + cot2θ.
Answer:
L.H.S
(sinθ + cosecθ)2 + (cosθ + secθ)2
= sin2θ + 2sinθcosecθ + cosec2θ + cos2θ + 2cosθsecθ + sec2θ
= sin2θ + 2 + cosec2θ + cos2θ + 2 + sec2θ
= sin2θ + cos2θ + sec2θ + cosec2θ + 2 + 2
= 1 + 2 + 2 + (1 + tan2θ) + (1 + cot2θ)
= 7 + tan2θ + cot2θ
Question 9.
Prove the following by using trigonometric identities:
2sec2θ — sec4θ — 2cosec2θ + cosec4θ = cot4θ — tan4θ.
Answer:
2sec2θ — sec4θ — 2cosec2θ + cosec4θ
= 2(1 + tan2θ) – (1 + tan2θ)2 – 2(1 + cot2θ) + (1 + cot2θ)2
open all the brackets and cancel terms
= cot4θ – tan4θ
Question 10.
Prove the following by using trigonometric identities:
(sinθ — secθ)2 + (cosθ — cosecθ)2 = (1 — secθ ⋅ cosecθ)2.
Answer:
L.H.S
= sin2θ – 2sinθsecθ + sec2θ + cos2θ – 2cosθcosecθ + cosec2θ
= (sin2θ + cos2θ) – (2sinθsecθ + 2cosθcosecθ) + cosec2θ + sec2θ
Question 11.
Prove the following by using trigonometric identities:
Answer:
L.H.S
Question 12.
Prove the following by using trigonometric identities:
Answer:
Question 13.
Prove the following by using trigonometric identities:
Answer:
L.H.S
Question 14.
Prove the following by using trigonometric identities:
Answer:
L.H.S
∵ tanθcotθ = 1
Question 15.
Prove the following by using trigonometric identities:
Answer:
∵
Question 16.
Prove the following by using trigonometric identities:
Answer:
Question 17.
Prove the following by using trigonometric identities:
sin4θ – cos4θ = sin2θ – cos2θ = 2sin2θ – 1 = 1 – 2 cos2θ.
Answer:
. . .(1)
Now,
Also,
By (1)
Question 18.
Prove the following by using trigonometric identities:
tan2A — tan2B =
Answer:
Question 19.
Prove the following by using trigonometric identities:
2(sin6θ + cos6θ) — 3(sin4θ + cos4θ) + 1 = 0
Answer:
2((sin2θ)3 + (cos2θ)3) — 3(sin4θ + cos4θ) + 1
(∵ a2 + b2 = (a + b)2–2ab)
⇒ 2[(sin2θ + cos2θ)(sin4θ –sin2θcos2θ + cos4θ)]–3[(sin2θ + cos2θ)–2sin2θcos2θ] + 1
= 2[(sin4θ –sin2θcos2θ + cos4θ)]–3[1–2sin2θcos2θ] + 1
= 2sin4θ–2sin2θcos2θ + 2cos4θ–3 + 6sin2θcos2θ + 1
= 2(sin4θ + 4sin2θcos2θ + cos4θ)–2
= 2(sin2θ + cos2θ)2–2
= 2–2
= 0
Question 20.
If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p2 — 1) = 2p.
Answer:
L.H.S
q(p2 — 1)
substituting p and q.
(sec + cosec
)((sin
+ cos
)2–1)
=
also
Question 21.
If tanθ + sin = a and tanθ — sinθ = b, then prove that a2 — b2 =
Answer:
tan + sin
= a [1]
tan — sin
= b [2]
adding and subtracting [1] and [2]
2tanθ = a + b and 2sinθ = a–b
L.H.S
a2 — b2 = (a + b)(a–b)
⇒ a2 — b2 = (2tanθ)( 2sinθ)
⇒ a2 — b2 = 4tanθsinθ
R.H.S
4√(ab)
⇒
=
L.H.S. = R.H.S
Question 22.
acosθ + bsinθ = p and asinθ— bcosθ = q, then prove that a2 + b2 = p2 + q2.
Answer:
acosθ + bsinθ = p (i)
asinθ – bcosθ = q (ii)
Squaring and adding (i) and (ii)
∴ (acosθ + bsinθ)2 + (asinθ – bcosθ)2 = p2 + q2
∴ a2cos2θ + 2abcosθsinθ + b2sin2θ + a2sin2θ – 2absinθcosθ + b2cos2θ = p2 + q2
∴ a2(cos2θ + sin2θ) + b2 (sin2θ + cos2θ) = p2 + q2
∴a2 (1) + b2 (1) = p2 + q2
∴ a2 + b2 = p2 + q2
Question 23.
secθ + tanθ = p, then obtain the values of secθ, tanθ and sinθ in terms of p.
Answer:
secθ + tanθ [1]
sec2θ –tan2θ = 1
⇒ (secθ – tanθ)( (secθ + tanθ)) = 1
⇒ (secθ – tanθ) p = 1
⇒ (secθ – tanθ) = [2]
Adding and subtracting [1] and [2]
and
⇒ and
⇒ and
Question 24.
Evaluate the following: ⋅ tan17 tan38 tan60 tan52 tan73 — 3(sin232 + sin258)
Answer:
Question 25.
Evaluate the following:
Answer:
Question 26.
If sinA + cosA = √2 sin(90—A), then obtain the value of cotA.
Answer:
sinA + cosA = sin(90—A)
dividing the complete equation by cosA
Question 27.
If cosecθ = √2, then find the value of
Answer:
cosecθ = √2
Now,
Question 28.
If then evaluate
Answer:
1 + tan2θ = sec2θ
⇒
and
Now,
Question 29.
If , 0 < θ < 90, find the value of sinθ and tanθ.
Answer:
sin2θ = 1 – cos2θ
∴
Question 30.
If θ is the measure of an acute angle such that bsinθ = acosθ, then is =
A.
B.
C.
D.
Answer:
dividing both numerator and denominator by cosθ.
Question 31.
Which of the following is correct for some 0 such that 0 ≤ θ < 90?
A. >1
B. = 1
C. sec θ = 0
D. <1
Answer:
for θ = 0, secθ = 1
⇒ option B is correct
Question 32.
If , then
is ….
A.
B.
C.
D. 3
Answer:
Question 33.
If , then the value of
is ____
A.
B.
C.
D.
Answer:
⇒
∵ tan2θ =
Question 34.
If , then the value of
is ____
A. 7
B.
C.
D.
Answer:
dividing numerator and denominator by sinθ
Question 35.
If cosecA = and A + B = 90, then secB is ………..
A.
B.
C.
D.
Answer:
∵ A = 90 – B
Question 36.
If θ is the measure of an acute angle and √3 sinθ = cosθ, then θ is ____
A. 30
B. 45
C. 60
D. 90
Answer:
Question 37.
If then the value of (sinA + cosA) secA is …..
A.
B.
C.
D.
Answer:
(sinA + cosA) secA
Question 38.
If then the value of
is ….
A.
B. 3
C.
D.
Answer:
0
sinA = cosA.tanA
Now,
Question 39.
In ΔABC, if m∠ABC = 90, m∠ACB = 45 and AC = 6, then area of ΔABC is …..
A. 18
B. 36
C. 9
D.
Answer:
given ∠B = 90, ∠C = 45
by angle sum property
⇒ ∠A = 45
⇒ AB = BC
also AB2 + BC2 = AC2 (by (Pythagoras theorem)
⇒ AB√2 = AC
∵ AC = 6
⇒ AB =
⇒ AB = BC = 3√2
Area of triangle =
⇒ Area =
= 9
Question 40.
If cos245 – cos230 = x ⋅ cos45 ⋅ sin45, then x is …….
A. 2
B.
C.
D.
Answer:
cos245 – cos230 = x ⋅ cos45 ⋅ sin45
Question 41.
If A and B are complementary angles, then sinA ⋅ secB is ____
A. 1
B. 0
C. —1
D. 2
Answer:
Question 42.
The value of tan20 tan25 tan45 tan65 tan70 is ………….
A. —1
B. 1
C. 0
D. √3
Answer:
tan20.tan25.tan45.tan65.tan70
= cot(90–20).cot(90–25).1.tan65tan70
= cot70.cot65.tan65.tan70
= 1
Question 43.
If 7θ and 2θ are measure of acute angles such that sin7θ = cos2θ then 2sin3θ — √3 tan3θ is ……….
A. 1
B. 0
C. —1
D. 1 — √3
Answer:
sin7θ = cos2θ
∴ sin7θ = sin(90–2θ )
⇒ 9θ = 90
⇒ θ = 10
2sin3θ –√3tan3θ
put θ = 10
2sin3×20 – √3×tan (10× 3)
= 0
Question 44.
If A + B = 90, then is ….
A. cot2B
B. tan2A
C. cot2A
D. —cot2A
Answer:
Question 45.
For ΔABC, sin = ……..
A. sin
B. sinA
C. cos
D. cosA
Answer:
Question 46.
= ………..
A. 1
B. 2
C. 3
D. 0
Answer:
Question 47.
If 7cos2θ + 3sin2θ = 4, then cotθ is ___
A. 7
B.
C.
D.
Answer:
7cos2 + 3sin2
= 4
⇒ 7(1–sin2θ) + 3sin2 = 4
⇒ 7–4sin2θ = 4
⇒ sinθ =
⇒ θ = 60
⇒ cot60 =
Question 48.
If tan5θ ⋅ tan4θ = 1, θ is ____
A. 7
B. 3
C. 10
D. 9
Answer:
tan5θ.cot(90–4θ) = 1
⇒ cot(90–4θ) = tan5θ.
⇒ 5θ = 90– 4θ
⇒ θ = 10
Question 49.
If A and B are measures of acute angles and tanA = and sinB =
, then cos(A + B) is ………………..
A. 0
B.
C.
D.
Answer:
tanA =
also tan30 =
⇒ A = 30
Sin B =
⇒ B = 30
⇒ A + B = 30 + 30 = 60
⇒ cos(A + B) = Cos(60)