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
From Worksheet on Trigonometric Identities to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Recent Articles
-
2nd Grade Geometry Worksheet | Plane and Solid Shapes | Point | Line
Dec 14, 24 02:12 PM
2nd grade geometry worksheet -
2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic
Dec 14, 24 12:25 PM
2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students. -
Patterns in Numbers | Patterns in Maths |Math Patterns|Series Patterns
Dec 13, 24 08:43 AM
We see so many patterns around us in our daily life. We know that a pattern is an arrangement of objects, colors, or numbers placed in a certain order. Some patterns neither grow nor reduce but only r… -
Patterns in Math | Missing Number | Counting Numbers | Worksheets
Dec 13, 24 12:31 AM
Finding patterns in math is very important to understand the sequence in the series. We need to find the exact missing number that from the group of numbers. The counting numbers may be counting -
Concept of Pattern | Similar Patterns in Mathematics | Similar Pattern
Dec 12, 24 11:22 PM
Concept of pattern will help us to learn the basic number patterns and table patterns. Animals such as all cows, all lions, all dogs and all other animals have dissimilar features. All mangoes have si…
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.