Worksheet on Trigonometric Identities
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 sin\(^{4}\) θ – cos\(^{4}\) θ = 2 sin\(^{2}\) θ – 1
3. Prove the trigonometric identity sin\(^{4}\) θ - cos\(^{4}\) θ + 1 = 2 sin\(^{2}\) θ
4. Prove the trigonometric identity cos\(^{4}\) θ - sin\(^{4}\) θ = 2 cos\(^{2}\) θ – 1
5. Prove the trigonometric identity sin α cos α(tan α - cot α) = 2 sin2 α - 1
6. Prove the trigonometric identity cos\(^{6}\) θ + sin\(^{6}\) θ = 1 - 3 sin\(^{2}\) θ ∙ cos\(^{2}\) θ
Hint: cos\(^{6}\) θ + sin\(^{6}\) θ = \((cos^{2} θ)^{3}\) + \((sin^{2} θ)^{3}\)
= (cos\(^{2}\) θ + sin\(^{2}\) θ)(cos\(^{4}\) θ - cos\(^{2}\) θ ∙ sin\(^{2}\) θ + sin\(^{4}\) θ)
= 1 ∙ {cos\(^{4}\) + sin\(^{4}\) θ - cos\(^{2}\) θ ∙ sin\(^{2}\) θ}
= 1 ∙ {\((cos^{2} θ + sin^{2} θ)^{2}\) - 2 cos\(^{2}\) θ ∙ sin\(^{2}\) θ - cos\(^{2}\) θ ∙ sin\(^{2}\) θ}
= 1 ∙ {\((cos^{2} θ + sin^{2} θ)^{2}\) - 3 cos\(^{2}\) θ ∙ sin\(^{2}\) θ}
7. Prove the trigonometric identity (a cos θ + b sin θ)\(^{2}\) + (a cos θ - b sin θ)\(^{2}\) = a\(^{2}\) + b\(^{2}\)
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 sec\(^{2}\) θ
10. Prove the trigonometric identity \(\frac{1}{sin^{2} A}\) - \(\frac{1}{sin^{2} B}\) = \(\frac{cos^{2} A - cos^{2} B}{sin^{2} A ∙ sin^{2} B}\)
11. Prove the trigonometric identity \(\frac{1}{1 + cos A}\) + \(\frac{1}{1 - cos A}\) = 2 csc\(^{2}\) A
12. Prove the trigonometric identity (cot θ + csc θ)2 = \(\frac{1 + cos θ}{1 - cos θ}\)
13. Prove the trigonometric identity \(\frac{1}{1 - sin A}\) - \(\frac{1}{1 + sin A}\) = 2 tan A ∙ sec A
14. Prove the trigonometric identity \(\frac{1}{1 - cos A}\) + \(\frac{1}{1 + cos A}\) = 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 \(\frac{cos A}{1 + sin A}\) + \(\frac{cos A}{1 - sin A}\) = 2 sec A
17. Prove the trigonometric identity \(\frac{1}{1 - sin A}\) + \(\frac{1}{1 + sin A}\) = 2 sec\(^{2}\) A
18. Prove the trigonometric identity \(\frac{1}{sin A + cos A}\) + \(\frac{1}{sin A - cos A}\) = \(\frac{2 sin A}{1 – cos^{2} A}\)
19. Prove the trigonometric identity \(\frac{1 + sin θ}{1 - sin θ}\) = (sec θ + tan θ)2
20. Prove the trigonometric identity \(\frac{1 – sin A}{cos A}\) = \(\frac{cos A}{1 + sin A}\)
21. Prove the trigonometric identity \(\frac{cos θ}{1 + sin θ}\) + \(\frac{1 + sin θ}{cos θ}\) = 2 sec θ
22. Prove the trigonometric identity \((\frac{1 + cos A}{sin A})^{2}\) = \(\frac{1 + cos A}{1 - cos A}\)
23. Prove the trigonometric identity \(\frac{sin A}{1 + cos A}\) + \(\frac{1 + cos A}{sin A}\) = 2 csc θ
24. Prove the trigonometric identity \(\sqrt{\frac{1 + sin θ}{1 - sin θ}}\) = sec θ + tan θ
25. Prove the trigonometric identity \(\sqrt{\frac{1 - cos A}{1 + cos A}}\) = csc A – cot A
26. Prove the trigonometric identity \(\sqrt{\frac{1 - cos θ}{1 + cos θ}}\) = \(\frac{sin θ}{1 + cos θ}\)
27. Prove the trigonometric identity \(\sqrt{\frac{1 - sin A}{1 + sin A}}\) = sec A – tan A
28. Prove the trigonometric identity \(\sqrt{\frac{csc A - 1}{csc A + 1}}\) = \(\sqrt{\frac{1 - sin A}{cos A}}\)
29. Prove the trigonometric identity \(\sqrt{\frac{1 + cos A}{1 - cos A}}\) = csc A + cot A
30. Prove the trigonometric identity \(\sqrt{\frac{1 + sin A}{1 - sin A}}\) + \(\sqrt{\frac{1 - sin A}{1 + sin A}}\) = 2 sec A
31. Prove the trigonometric identity (1 + cos θ)(1 – cos θ)(1 + cot\(^{2}\) θ) = 1
32. Prove the trigonometric identity (1 + tan\(^{2}\) A) sin A ∙ cos A = tan A
33. Prove the trigonometric identity cot\(^{2}\) α + cot\(^{2}\) β = \(\frac{sin^{2} β - sin^{2} α}{sin^{2} α ∙ sin^{2} β}\)
34. Prove the trigonometric identity tan A + cot A = sec A ∙ csc A
35. Prove the trigonometric identity \(\frac{csc A}{tan A + cot A}\) = cos A
35. Prove the trigonometric identity sec\(^{2}\) θ + csc\(^{2}\) θ = sec\(^{2}\) θ ∙ csc\(^{2}\) θ
36. Prove the trigonometric identity tan\(^{2}\) θ + cot\(^{2}\) θ + 2 = sec\(^{2}\) θ ∙ csc\(^{2}\) θ
37. Prove the trigonometric identity tan\(^{4}\) θ + tan\(^{2}\) θ = sec\(^{4}\) θ - sec\(^{2}\) θ
38. Prove the trigonometric identity csc\(^{4}\) θ – 2 csc\(^{2}\) θ + 2 sec\(^{2}\) θ - sec\(^{4}\) θ = cot\(^{4}\) θ - tan\(^{4}\) θ.
Hint: (csc\(^{4}\) θ – 2 csc\(^{2}\) θ) - (sec\(^{4}\) θ - 2 sec\(^{2}\) θ)
= (csc\(^{4}\) θ – 2 csc\(^{2}\) θ + 1 - 1) - (sec\(^{4}\) θ - 2 sec\(^{2}\) θ + 1 - 1)
= (csc\(^{4}\) θ – 2 csc\(^{2}\) θ + 1) - 1 - (sec\(^{4}\) θ - 2 sec\(^{2}\) θ + 1) + 1
= (csc2 θ - 1)2 - (sec2 θ - 1)2
= (cot2 θ)2 - (tan2 θ)2
39. Prove the trigonometric identity \(\frac{sin A – 2 sin^{3} A}{2cos^{3} A – cos A}\) = tan A.
40. Prove the trigonometric identity \(\frac{cos θ}{csc θ + 1}\) + \(\frac{cos θ}{csc θ - 1}\) = 2 tan θ
41. Prove the trigonometric identity \(\frac{cos θ}{1 - tan θ}\) + \(\frac{sin θ}{1 - cot θ}\) = sin θ + cos θ
42. Prove the trigonometric identity
\(\frac{1}{sec θ - tan θ}\) - \(\frac{1}{cos θ}\) = \(\frac{1}{cos θ}\) - \(\frac{1}{sec θ + tan θ}\)
Hint: \(\frac{1}{sec θ - tan θ}\) + \(\frac{1}{sec θ + tan θ}\) = \(\frac{2}{cos θ}\)
43. Prove the trigonometric identity \(\frac{tan θ}{csc θ + 1}\) + \(\frac{tan θ}{csc θ - 1}\) = 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 \(\frac{tan A + cot B}{cot A + tan B}\) = \(\frac{tan A}{tan B}\)
46. Prove the trigonometric identity \(\frac{tan A + sec A - 1}{tan A – sec A + 1}\) = \(\frac{1 + sin A}{cos A}\)
Hint: \(\frac{tan A + sec A - 1}{tan A – sec A + 1}\)
= \(\frac{tan A + sec A - 1}{tan A – sec A + 1}\) ∙ \(\frac{tan A + sec A + 1}{tan A – sec A + 1}\)
= \(\frac{(tan A + sec A)^{2} - 1}{(tan A + 1)^{2} – sec^{2} A}\)
47. Prove the trigonometric identity \(\frac{1 + sin α}{csc α – cot α}\) - \(\frac{1 - sin α}{csc α + cot α}\) = 2 (1 + cot α)
48. Prove the trigonometric identity \(\frac{1}{cos θ + sin θ - 1}\) + \(\frac{1}{cos θ + sin θ + 1}\) = sec θ + csc θ
49. Prove the trigonometric identity \(\frac{tan A}{1 - cot A}\) + \(\frac{cot A}{1 - tan A}\) = 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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