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 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

Worksheet on Trigonometric Identities

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 = cos2Acos2Bsin2Asin2B

11. Prove the trigonometric identity 11+cosA + 11cosA = 2 csc2 A

12. Prove the trigonometric identity (cot θ + csc θ)21+cosθ1cosθ

13. Prove the trigonometric identity 11sinA - 11+sinA = 2 tan A ∙ sec A

14. Prove the trigonometric identity 11cosA + 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 + cosA1sinA = 2 sec A

17. Prove the trigonometric identity 11sinA + 11+sinA = 2 sec2 A

18. Prove the trigonometric identity 1sinA+cosA + 1sinAcosA = 2sinA1cos2A

19. Prove the trigonometric identity 1+sinθ1sinθ = (sec θ + tan θ)2

20. Prove the trigonometric identity 1sinAcosA = cosA1+sinA

21. Prove the trigonometric identity cosθ1+sinθ + 1+sinθcosθ = 2 sec θ

22. Prove the trigonometric identity (1+cosAsinA)2 = 1+cosA1cosA

23. Prove the trigonometric identity sinA1+cosA + 1+cosAsinA = 2 csc θ

24. Prove the trigonometric identity 1+sinθ1sinθ = sec θ + tan θ

25. Prove the trigonometric identity 1cosA1+cosA = csc A – cot A

26. Prove the trigonometric identity 1cosθ1+cosθ = sinθ1+cosθ

27. Prove the trigonometric identity 1sinA1+sinA = sec A – tan A

28. Prove the trigonometric identity cscA1cscA+1 = 1sinAcosA

29. Prove the trigonometric identity 1+cosA1cosA = csc A + cot A

30. Prove the trigonometric identity 1+sinA1sinA + 1sinA1+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 = sin2βsin2αsin2αsin2β

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 sinA2sin3A2cos3AcosA = tan A.

40. Prove the trigonometric identity cosθcscθ+1 + cosθcscθ1 = 2 tan θ

41. Prove the trigonometric identity  cosθ1tanθ + sinθ1cotθ = sin θ + cos θ

42. Prove the trigonometric identity 

                       1secθtanθ - 1cosθ = 1cosθ - 1secθ+tanθ

Hint: 1secθtanθ + 1secθ+tanθ2cosθ


43. Prove the trigonometric identity tanθcscθ+1 + tanθcscθ1 = 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+secA1tanAsecA+1 = 1+sinAcosA

Hint: tanA+secA1tanAsecA+1

     = tanA+secA1tanAsecA+1 ∙ tanA+secA+1tanAsecA+1

     = (tanA+secA)21(tanA+1)2sec2A


47. Prove the trigonometric identity 1+sinαcscαcotα - 1sinαcscα+cotα = 2 (1 + cot α)

48. Prove the trigonometric identity 1cosθ+sinθ1 + 1cosθ+sinθ+1 = sec θ  + csc θ

49. Prove the trigonometric identity tanA1cotA + cotA1tanA = 1 + sec A ∙ csc A

50. Prove the trigonometric identity (sec x - 1)2 - (tan x - sin x)2 = (1 - cos x)2




10th Grade Math

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