14. Trigonometrical IdentitiesΒΆ

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 \(\sin 3A\sin(B - C) + \sin 3B\sin(C - A) + \sin3C\sin(A - B) = 0\)