# 14. Trigonometrical Identities¶

In this chapter we will have only problems and we will use the theory we have learned till now.

1. If $A + B + C = \pi,$ prove that $\sin^2A + \sin^2B - \sin^2C = 2\sin A\sin B\sin C$

2. If $A + B + C = 180^\circ,$ prove that $\sin^2\frac{A}{2} + \sin^2\frac{B}{2} + \sin^2\frac{C}{2} = 1 - 2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

3. Show that $\sin^2A + \sin^2B + 2\sin A\sin B\cos(A + B) = \sin^2(A + B)$

4. If $A + B + C = 180^\circ,$ prove that $\cos^2A + \cos^2B + \cos^2C + 2\cos A\cos B\cos C = 1$

5. If $A + B + C = 180^\circ,$ prove that $\sin^2A + \sin^2B + \sin^2C = 2(1 + \cos A \cos B \cos C)$

6. If $A + B + C = 180^\circ,$ prove that $\cos^2A + \cos^2B - \cos^2C = 1 - 2\sin A\sin B\sin C$

7. If $A + B + C = 180^\circ,$ prove that $\cos^2\frac{A}{2} + \cos^2\frac{B}{2} - \cos^2\frac{C}{2} = 2\cos\frac{A}{2}\cos\frac{B}{2}\sin\frac{C}{2}$

8. If $A + B + C = 180^\circ,$ prove that $\cos^2\frac{A}{2} + \cos^2\frac{B}{2} + \cos^2\frac{C}{2} = 2 + 2\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$

9. If $A + B + C = \frac{\pi}{2},$ prove that $\sin^2A + \sin^2B + \sin^2C = 1 - 2\sin A\sin B\sin C$

10. If $A + B + C = \frac{\pi}{2},$ prove that $\cos^2A + \cos^2B + \cos^2C = 2 + 2\sin A\sin B\sin C$

11. If $A + B + C = 2\pi,$ prove that $\cos^2A + \cos^2B + \cos^2C - 2\cos A\cos B\cos C = 1$

12. If $A + B = C,$ prove that $\cos^2A + \cos^2B + \cos^2C - 2\cos A\cos B\cos C = 1$

13. If $A + B = \frac{\pi}{3},$ prove that $\cos^2A + \cos^2B - \cos A\cos B = \frac{3}{4}$

14. Show that $\cos^2B + \cos^2(A + B) - 2\cos A\cos B\cos(A + B)$ is independent of $B.$

15. If $A + B + C = \pi$ and $A + B = 2C,$ prove that $4(\sin^2A + \sin^2B - \sin A\sin B) = 3$

16. If $A + B + C = 2\pi,$ prove that $\cos^2B + \cos^2C - \sin^2A - 2\cos A\cos B\cos C = 0$

17. If $A + B + C = 0,$ prove that $\cos^2A + \cos^2B + \cos^2C = 1 + 2\cos A\cos B\cos C$

18. Prove that $\cos^2(B - C) + \cos^2(C - A) + \cos^2(A - B) = 1 + 2\cos(B - C)\cos(C - A)\cos(A - B)$

19. If $A + B + C = \pi,$ prove that $\sin A\cos B\cos C + \sin B\cos C\cos A + \sin C\cos A\cos B= \sin A\sin B\sin C$

20. If $A + B + C = \pi,$ prove that $\tan A + \tan B + \tan C = \tan A\tan B\tan C$

21. If $A + B + C = \pi,$ prove that $\tan\frac{A}{2}\tan\frac{B}{2} + \tan\frac{B}{2}\tan\frac{C}{2} + \tan\frac{C}{2}\tan\frac{A}{2} = 1$

22. If $A + B + C = \pi,$ prove that $\tan(B + C - A) + \tan(C + A - B) + \tan(A + B - C) = \tan(B + C - A)\tan(C + A - B)\tan(A + B - C)$

23. If $A + B + C = \pi,$ prove that $\cot B\cot C + \cot C\cot A + \cot A\cot B = 1$

24. In a $\triangle ABC,$ if $\cot A + \cot B + \cot C = \sqrt{3},$ prove that the triangle is equilateral.

25. If $A, B, C, D$ are angles of a quadrilateral, prove that $\frac{\tan A + \tan B + \tan C + \tan D}{\cot A + \cot B + \cot C + \cot D} = \tan A\tan B\tan C\tan D$

26. If $A + B + C = \frac{\pi}{2},$ show that $\cot A + \cot B + \cot C = \cot A\cot B\cot C$

27. If $A + B + C = \frac{\pi}{2},$ show that $\tan A\tan B + \tan B\tan C + \tan C\tan A = 1$

28. If $A + B + C = \pi,$ prove that $\tan 3A + \tan 3B + \tan 3C = \tan 3A\tan 3B\tan 3C$

29. If $A + B + C = \pi,$ prove that $\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2} = \cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$

30. If $A + B + C = \pi,$ prove that $\frac{\cot A + \cot B}{\tan A + \tan B} + \frac{\cot B + \cot C}{\tan B + \tan C} + \frac{\cot C + \cot A}{\tan C + \tan A} = 1$

31. Prove that $\tan(A - B) + \tan(B - C) + \tan(C - A) = \tan(A - B)\tan(B - C)\tan(C - A)$

32. If $x + y + z = 0,$ show that $\cot(x + y - z)\cot(z + x - y) + \cot(x + y - z)\cot(y + z - x) + \cot(y + z - x)\cot(z + x - y) = 1$

33. If $A + B + C = n\pi(n$ being zero or an integer $),$ show that $\tan A + \tan B + \tan C = \tan A\tan B\tan C$

34. If $A + B + C = \pi,$ prove that $\sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C$

35. If $A + B + C = \pi,$ prove that $\cos A + \cos B + \cos C - 1 = 4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

36. Prove that $\frac{\sin 2A + \sin 2B + \sin 2C}{\cos A + \cos B + \cos C - 1} = 8\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}$

37. If $A + B + C = \pi,$ prove that $\cos\frac{A}{2} + \cos\frac{B}{2} + \cos\frac{C}{2} = 4\cos\frac{\pi - A}{4}\cos\frac{\pi - B}{4}\cos\frac{\pi - C}{4}$

38. If $A + B + C = \pi,$ prove that $\sin \frac{A}{2} + \sin \frac{B}{2} + \sin \frac{C}{2} = 1 + 4\sin \frac{B + C}{4}\sin \frac{C + A}{4}\sin \frac{A + B}{4}$

39. If $A + B + C = \pi,$ prove that $\sin^2\frac{A}{2} + \sin^2\frac{B}{2} - \sin^2\frac{C}{2} = 1 - 2\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$

40. Prove that $1 + \cos 56^\circ + \cos 58^\circ - \cos 66^\circ = 4\cos28^\circ\cos29^\circ\sin 33^\circ$

41. If $A + B + C = \pi,$ prove that $\cos 2A + \cos 2B - \cos 2C = 1 - 4\sin A\sin B\cos C$

42. If $A + B + C = \pi,$ prove that $\sin 2A + \sin 2B - \sin 2C = 4\cos A\cos B\sin C$

43. If $A + B + C = \pi,$ prove that $\sin A + \sin B + \sin C = 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$

44. If $A + B + C = \pi,$ prove that $\cos A + \cos B - \cos C = 4\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2} - 1$

45. If $A + B + C = \pi,$ prove that $\sin(B + C - A) + \sin(C + A - B) + \sin(A + B - C) = 4\sin A\sin B\sin C$

46. If $A + B + C = \pi,$ prove that $\frac{\cos A}{\sin B\sin C} + \frac{\cos B}{\sin C\sin A} + \frac{\cos C}{\sin A\sin B} = 2$

47. If $A + B + C = \pi,$ prove that $\frac{\sin 2A + \sin 2B + \sin 2C}{\sin A + \sin B + \sin C} = 8\sin \frac{A}{2}\ \sin \frac{B}{2}\sin \frac{C}{2}$

48. If $x + y + z = \frac{\pi}{2},$ prove that $\cos(x - y - z) + \cos(y - z - x) + \cos(z - x - y) - 4\cos x\cos y\cos z = 0$

49. Show that $\sin(x - y) + \sin(y - z) + \sin(z - x) + 4\sin\frac{x - y}{2}\sin\frac{y - z}{2}\sin \frac{z - x}{2} = 0$

50. If $A + B + C = 180^\circ,$ prove that $\sin(B + 2C) + \sin(C + 2A) + \sin(A + 2B) = 4\sin\frac{B - C}{2} \sin\frac{C - A}{2}\sin\frac{A - B}{2}$

51. If $A + B + C = \pi,$ prove that $\sin\frac{B + C}{2} + \sin \frac{C + A}{2} + \sin \frac{A + B}{2} = 4\cos\frac{\pi - A}{4}\cos\frac{\pi - B}{4}\cos\frac{\pi - C}{4}$

52. If $xy + yz + zx = 1,$ prove that $\frac{x}{1 - x^2} + \frac{y}{1 - y^2} + \frac{z}{1 - z^2} = \frac{4xyz}{(1 - x^2)(1 - y^2)(1 - z^2)}$

53. If $x + y + z = xyz,$ show that $\frac{3x - x^3}{1 - 3x^2} + \frac{3y - y^3}{1 - 3y^2} + \frac{3z - z^3}{1 - 3z^2} = \frac{3x - x^3}{1 - 3x^2}.\frac{3y - y^3}{1 - 3y^2}.\frac{3z - z^3}{1 - 3z^2}$

54. If $x + y + z = xyz,$ prove that $\frac{2x}{1 - x^2} + \frac{2y}{1 - y^2} + \frac{2z}{1 - z^2} = \frac{2x}{1 - x^2}.\frac{2y}{1 - y^2}.\frac{2z}{1 - z^2}$

55. If $x + y + z = xyz,$ prove that $x(1 - y^2)(1 - z^2) + y(1 - z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) = 4xyz$

56. If $A + B + C + D = 2\pi,$ prove that $\cos A + \cos B + \cos C + \cos D = 4\cos\frac{A + B}{2}\cos\frac{B + C}{2}\cos\frac{C + A}{2}$

57. If $A + B + C = 2S,$ prove that $\cos^2S + \cos^2(S - A) + \cos^2(S - B) + \cos^2(S - C) = 2 + 2\cos A\cos B\cos C$

58. If $A + B + C = \pi,$ prove that $\tan^2\frac{A}{2} + \tan^2\frac{B}{2} + \tan^2\frac{C}{2}\geq 1$

59. If $A + B + C = \pi,$ prove that $(\tan A + \tan B + \tan C)(\cot A + \cot B + \cot C) = 1 + \sec A\sec B\sec C$

60. If $A + B + C = \pi,$ prove that $(\cot B + \cot C)(\cot C + \cot A)(\cot A + \cot C) = \cosec A\cosec B\cosec C$

61. If $A + B + C = \pi,$ prove that $\frac{1}{2}\sum \sin^2A(\sin 2B + \sin 2C) = 3\sin A\sin B\sin C$

62. If $A + B + C + D = 2\pi,$ prove that $\cos A - \cos B + \cos C - \cos D = 4\sin\frac{A + B}{2}\sin\frac{A + D}{2}\cos \frac{A + C}{2}$

63. If $A, B, C, D$ be the angles of a cyclic quadrilateral, prove that $\cos A + \cos B + \cos C + \cos D = 0$

64. If $A + B + C = \pi,$ prove that $\cot^2A + \cot^2B + \cot^2C \geq 1$

65. If $A + B + C = \pi,$ prove that $\cos \frac{A}{2}\cos\frac{B - C}{2} + \cos\frac{B}{2}\cos\frac{C - A}{2} + \cos \frac{C}{2}\cos\frac{A -B}{2} = \sin A + \sin B + \sin C$

66. In a $\triangle ABC,$ prove that $\sin 3A\sin(B - C) + \sin 3B\sin(C - A) + \sin3C\sin(A - B) = 0$