# 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 $\inline \dpi{120} \frac { 16 \cos A + 2 \sin A } { 24 \cos A + 2 \sin A }$
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 :$\inline \dpi{120} \tan \left( \frac { \angle A + \angle B } { 2 } \right) = \cot \frac { \angle C } { 2 }$
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:
$\dpi{120} \frac { \sin 70 ^ { \circ } } { \cos 20 ^ { \circ } } + \frac { \csc 36 ^ { \circ } } { \sec 54 ^ { \circ } } - \frac { 2 \cos 43 ^ { \circ } \csc 47 ^ { \circ } } { \tan 10 ^ { \circ } \tan 40 ^ { \circ } \tan 50 ^ { \circ } \tan 80 ^ { \circ } }$
11. Evaluate:
$\inline \dpi{120} \frac { 2 } { 3 } \csc ^ { 2 } 58 ^ { \circ } - \frac { 2 } { 3 } \cot 58 ^ { \circ }\cdot \tan 32 ^ { \circ } - \frac { 5 } { 3 } \tan 13 ^ { \circ }\cdot\tan 37 ^ { \circ } \cdot \tan 45 ^ { \circ } \cdot \tan 53 ^ { \circ } \cdot \tan 77 ^ { \circ }$
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 = $\inline \dpi{120} \sqrt { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } }$
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 = $\inline \dpi{120} \frac{\sqrt{2}}{4}$
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 $\inline \dpi{120} \sqrt { \cos A \csc B - \cos A \sin B } = \sin A$
28. If cos 2 θ – sin 2 θ = tan 2 φ , prove that $\inline \dpi{120} \cos \phi = \frac { 1 } { \sqrt { 2 } \cos \theta }$ .
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  $\inline \dpi{120} \cos ^ { 2 } \alpha = \frac { a ^ { 2 } - 1 } { b ^ { 2 } - 1 }$
31. If  $\inline \dpi{120} \frac { \cos \alpha } { \cos \beta } = m \text { and } \frac { \sin \alpha } { \sin \beta } = n$, prove that $\inline \dpi{120} \left( n ^ { 2 } - m ^ { 2 } \right) \sin ^ { 2 } \beta = 1 - m ^ { 2 }$.
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 $\inline \dpi{120} \frac{5}{x}$ = tan θ fnd the value of $\inline \dpi{120} 5 \left( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } \right)$
36. If tan θ + sin θ = m and tan θ – sin θ = n, prove that $\inline \dpi{120} m ^ { 2 } - n ^ { 2 } = 4 \sqrt { m n }$
37. If A + B = 90°, then prove that
$\dpi{120} \sqrt { \frac { \tan A \tan B + \tan A \cot B } { \sin A \sec B } - \frac { \sin ^ { 2 } B } { \cos ^ { 2 } A } } = \tan A$
38. Prove that (sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tan2 θ + cot2 θ.
39. Prove:
$\dpi{120} \frac { \cot A - \cos A } { \cot A + \cos A } = \frac { \csc A - 1 } { \csc A + 1 }$
40. Prove:
$\dpi{120} \sqrt { \frac { 1 - \sin A } { 1 + \sin A } } = \sec A - \tan A$
41. Prove:
$\dpi{120} \frac { 1 + \tan ^ { 2 } A } { 1 + \cot ^ { 2 } A } = \left( \frac { 1 - \tan A } { 1 - \cot A } \right) ^ { 2 } = \tan ^ { 2 } A$
42. Prove:
$\dpi{120} \frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } + \frac { \sin \theta + \cos \theta } { \sin \theta - \cos \theta } = \frac { 2 } { 2 \sin ^ { 2 } \theta - 1 }$
43. Prove:
$\dpi{120} \frac { \tan \theta } { 1 - \cot \theta } + \frac { \cot \theta } { 1 - \tan \theta } = 1 + \tan \theta + \cot \theta$
44. Prove:
$\dpi{120} \frac { \cot ^ { 2 } A ( \sec A - 1 ) } { 1 + \sin A } = \sec ^ { 2 } A \left( \frac { 1 - \sin A } { 1 + \sec A } \right)$
45. If sec θ + tan θ = p, show that $\inline \dpi{120} \frac { p ^ { 2 } - 1 } { p ^ { 2 } + 1 } = \sin \theta$
46. Prove:
$\dpi{120} \frac { \cos A } { 1 - \sin A } + \frac { 1 - \sin A } { \cos A } = 2 \sec A$
47. Prove:
$\dpi{120} \frac { \sin A + \cos A } { \sin A - \cos A } + \frac { \sin A - \cos A } { \sin A + \cos A } = \frac { 2 } { \sin ^ { 2 } A - \cos ^ { 2 } A }$
48. Prove:
$\dpi{120} \frac { 1 } { \sec \theta - \tan \theta } - \frac { 1 } { \cos \theta } = \frac { 1 } { \cos \theta } - \frac { 1 } { \sec \theta + \tan \theta }$
49. Prove:
$\dpi{120} \frac { \tan \theta + \sec \theta - 1 } { \tan \theta - \sec \theta + 1 } = \frac { 1 + \sin \theta } { \cos \theta }$
50. Prove:
$\dpi{120} \frac { \cos A } { 1 - \tan A } + \frac { \sin A } { 1 - \cot A } = \sin A + \cos A$
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:
$\dpi{120} \frac { \sin \theta } { ( \cot \theta + \csc \theta ) } = 2 + \frac { \sin \theta } { ( \cot \theta - \csc \theta ) }$
55. Prove:
$\dpi{120} \frac { \csc \theta + \cot \theta } { \csc \theta + \cot \theta } = ( \csc \theta + \cot \theta ) ^ { 2 }= 1 + 2 \cot ^ { 2 } \theta + 2 \csc \theta \cot \theta$