In worksheet on trigonometric identities we will prove various types of practice questions on establishing identities. Here you will get 50 different types of proving trigonometric identities questions with some selected questions hints.
1. Prove the trigonometric identity sin θ cos θ (tan θ + cot θ) = 1.
2. Prove the trigonometric identity sin4 θ – cos4 θ = 2 sin2 θ – 1
3. Prove the trigonometric identity sin4 θ - cos4 θ + 1 = 2 sin2 θ
4. Prove the trigonometric identity cos4 θ - sin4 θ = 2 cos2 θ – 1
5. Prove the trigonometric identity sin α cos α(tan α - cot α) = 2 sin2 α - 1
6. Prove the trigonometric identity cos6 θ + sin6 θ = 1 - 3 sin2 θ ∙ cos2 θ
Hint: cos6 θ + sin6 θ = (cos2θ)3 + (sin2θ)3
= (cos2 θ + sin2 θ)(cos4 θ - cos2 θ ∙ sin2 θ + sin4 θ)
= 1 ∙ {cos4 + sin4 θ - cos2 θ ∙ sin2 θ}
= 1 ∙ {(cos2θ+sin2θ)2 - 2 cos2 θ ∙ sin2 θ - cos2 θ ∙ sin2 θ}
= 1 ∙ {(cos2θ+sin2θ)2 - 3 cos2 θ ∙ sin2 θ}
7. Prove the trigonometric identity (a cos θ + b sin θ)2 + (a cos θ - b sin θ)2 = a2 + b2
8. Prove the trigonometric identity (cos A + sin A)2 + (cos A - sin A)2 = 2
9. Prove the trigonometric identity (1 + tan θ)2 + (1 - tan θ)2 = 2 sec2 θ
10. Prove the trigonometric identity 1sin2A - 1sin2B = cos2A−cos2Bsin2A∙sin2B
11. Prove the trigonometric identity 11+cosA + 11−cosA = 2 csc2 A
12. Prove the trigonometric identity (cot θ + csc θ)2 = 1+cosθ1−cosθ
13. Prove the trigonometric identity 11−sinA - 11+sinA = 2 tan A ∙ sec A
14. Prove the trigonometric identity 11−cosA + 11+cosA = 2 cot A ∙ csc A
15. Prove the trigonometric identity (1 + sec A + tan A)(1 - csc A + cot A) = 2
16. Prove the trigonometric identity cosA1+sinA + cosA1−sinA = 2 sec A
17. Prove the trigonometric identity 11−sinA + 11+sinA = 2 sec2 A
18. Prove the trigonometric identity 1sinA+cosA + 1sinA−cosA = 2sinA1–cos2A
19. Prove the trigonometric identity 1+sinθ1−sinθ = (sec θ + tan θ)2
20. Prove the trigonometric identity 1–sinAcosA = cosA1+sinA
21. Prove the trigonometric identity cosθ1+sinθ + 1+sinθcosθ = 2 sec θ
22. Prove the trigonometric identity (1+cosAsinA)2 = 1+cosA1−cosA
23. Prove the trigonometric identity sinA1+cosA + 1+cosAsinA = 2 csc θ
24. Prove the trigonometric identity √1+sinθ1−sinθ = sec θ + tan θ
25. Prove the trigonometric identity √1−cosA1+cosA = csc A – cot A
26. Prove the trigonometric identity √1−cosθ1+cosθ = sinθ1+cosθ
27. Prove the trigonometric identity √1−sinA1+sinA = sec A – tan A
28. Prove the trigonometric identity √cscA−1cscA+1 = √1−sinAcosA
29. Prove the trigonometric identity √1+cosA1−cosA = csc A + cot A
30. Prove the trigonometric identity √1+sinA1−sinA + √1−sinA1+sinA = 2 sec A
31. Prove the trigonometric identity (1 + cos θ)(1 – cos θ)(1 + cot2 θ) = 1
32. Prove the trigonometric identity (1 + tan2 A) sin A ∙ cos A = tan A
33. Prove the trigonometric identity cot2 α + cot2 β = sin2β−sin2αsin2α∙sin2β
34. Prove the trigonometric identity tan A + cot A = sec A ∙ csc A
35. Prove the trigonometric identity cscAtanA+cotA = cos A
35. Prove the trigonometric identity sec2 θ + csc2 θ = sec2 θ ∙ csc2 θ
36. Prove the trigonometric identity tan2 θ + cot2 θ + 2 = sec2 θ ∙ csc2 θ
37. Prove the trigonometric identity tan4 θ + tan2 θ = sec4 θ - sec2 θ
38. Prove the trigonometric identity csc4 θ – 2 csc2 θ + 2 sec2 θ - sec4 θ = cot4 θ - tan4 θ.
Hint: (csc4 θ – 2 csc2 θ) - (sec4 θ - 2 sec2 θ)
= (csc4 θ – 2 csc2 θ + 1 - 1) - (sec4 θ - 2 sec2 θ + 1 - 1)
= (csc4 θ – 2 csc2 θ + 1) - 1 - (sec4 θ - 2 sec2 θ + 1) + 1
= (csc2 θ - 1)2 - (sec2 θ - 1)2
= (cot2 θ)2 - (tan2 θ)2
39. Prove the trigonometric identity sinA–2sin3A2cos3A–cosA = tan A.
40. Prove the trigonometric identity cosθcscθ+1 + cosθcscθ−1 = 2 tan θ
41. Prove the trigonometric identity cosθ1−tanθ + sinθ1−cotθ = sin θ + cos θ
42. Prove the trigonometric identity
1secθ−tanθ - 1cosθ = 1cosθ - 1secθ+tanθ
Hint: 1secθ−tanθ + 1secθ+tanθ = 2cosθ
43. Prove the trigonometric identity tanθcscθ+1 + tanθcscθ−1 = 2 csc θ
44. Prove the trigonometric identity (sec θ + tan θ – 1)(sec θ - tan θ + 1) = 2 tan θ
Hint: (sec θ + tan θ – 1)(sec θ - tan θ + 1)
= [sec θ + (tan θ – 1)][sec θ - (tan θ - 1)]
= sec2 θ - (tan θ – 1)2
= sec2 θ - tan2 θ – 2 tan θ + 1
= (sec2 θ - tan2 θ) – 2 tan θ + 1
45. Prove the trigonometric identity tanA+cotBcotA+tanB = tanAtanB
46. Prove the trigonometric identity tanA+secA−1tanA–secA+1 = 1+sinAcosA
Hint: tanA+secA−1tanA–secA+1
= tanA+secA−1tanA–secA+1 ∙ tanA+secA+1tanA–secA+1
= (tanA+secA)2−1(tanA+1)2–sec2A
47. Prove the trigonometric identity 1+sinαcscα–cotα - 1−sinαcscα+cotα = 2 (1 + cot α)
48. Prove the trigonometric identity 1cosθ+sinθ−1 + 1cosθ+sinθ+1 = sec θ + csc θ
49. Prove the trigonometric identity tanA1−cotA + cotA1−tanA = 1 + sec A ∙ csc A
50. Prove the trigonometric identity (sec x - 1)2 - (tan x - sin x)2 = (1 - cos x)2
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