 # Problems in Introduction in Trigonometry

1. Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.
2. If tan A = cot B, prove that A + B = 90°.
3. If 8 cot A = 15, find
4. ∆ABC is right angled at B and ∆PQR is right angled at Q. If cos A = cos P, show that ∠A = ∠P.
5. If sec 4A = cosec (A – 20°), where 4A is an acute angle, fnd the value of A.
6. If A, B and C are interior angles of ∆ABC, then show that :
7. ∆ABC is right angled at B, BC =7 cm and AC – AB = 1 cm. Find the value of cos A – sin A.
8. Evaluate : sin 15° cos 75° + cos 15° sin 75°
9. In the figure, ABCD is a rectangle in which segments AP and AQ are drawn. Find the length (AP + AQ). 10. Evaluate:
11. Evaluate:
12. Find the value of sin5° + sin2 10° + sin2 80° +sin2 85°.
13. Prove that sin6θ + cos6θ = 3sin2θcos2θ.
14. Prove that : (cosec A – sin A)(sec A – cos A) (tan A + cot A) = 1
15. In an acute angled triangle ABC, if sin 2(A + B – C) = 1 and tan (B + C – A) = 3 , find the values of A, B and C.
16. If sin θ + cos θ = p and sec θ + cosec θ = q then prove that q(p2 – 1) = 2p.
17. If 2cosθ – sinθ = x and cosθ – 3sinθ = y, prove that 2x2 + y2 – 2xy = 5.
18. If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ =
19. Prove that : (1 + tan A tan B)2 + (tan A – tan B)2= sec2A·sec2B.
20. If x = r sin A cos C, y = r sin A sin C, z = r cosA, prove that r2 = x2 + y2 + z2 .
21. If tan A = √2 – 1 show that sinA·cosA =
22. If 1 + sin2 θ = 3 sin θ cos θ, then, prove that tan θ = 1 or 1/2.
23. If cosec θ – sin θ = l and sec θ – cos θ = m, show that l2m2 (l2 + m2 + 3) = 1.
24. If sin θ + cos θ = 1, prove that (cos θ – sin θ) = ± 1
25. If sin θ + sin2 θ + sin3 θ = 1, then prove that cos6θ – 4 cos4 θ + 8 cos2 θ = 4.
26. If tan2 θ = 1 + 2 tan2 φ, prove that 2 sin2 θ = 1 + sin2 φ .
27. If A + B = 90°, show that
28. If cos 2 θ – sin 2 θ = tan 2 φ , prove that  .
29. If x = a sec θ + b tan θ , y  = a tan θ + b sec θ prove that x2 – y2 = a2 – b2.
30. If sin α = a sin β and tan α = b tan β, then prove that
31. If  , prove that .
32. Prove that : (1 + cot A + tan A)(sin A – cos A) = sin A·tan A – cot A·cos A.
33. Prove that : (1 + cot A – cosec A)(1 + tan A + sec A) = 2.
34. Prove that 2 sec2 θ – sec4 θ – 2 cosec2 θ+ cosec4 θ = cot4 θ – tan4 θ.
35. If 5x = sec θ and  = tan θ fnd the value of
36. If tan θ + sin θ = m and tan θ – sin θ = n, prove that
37. If A + B = 90°, then prove that
38. Prove that (sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tan2 θ + cot2 θ.
39. Prove:
40. Prove:
41. Prove:
42. Prove:
43. Prove:
44. Prove:
45. If sec θ + tan θ = p, show that
46. Prove:
47. Prove:
48. Prove:
49. Prove:
50. Prove:
51. Prove that 2(sin6θ + cos6θ) – 3(sin4θ + cos4θ) + 1 = 0
52. Prove that sin6 A + cos6 A + 3 sin2 A cos2 A = 1.
53. (sin4 A – cos4 A + 1) cosec2 A = 2.
54. Prove:
55. Prove: