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=π,A + B + C = \pi, prove that sin⁡2A+sin⁡2B−sin⁡2C=2sin⁡Asin⁡Bsin⁡C\sin^2A + \sin^2B - \sin^2C = 2\sin A\sin B\sin C

  2. If A+B+C=180∘,A + B + C = 180^\circ, prove that sin⁡2A2+sin⁡2B2+sin⁡2C2=1−2sin⁡A2sin⁡B2sin⁡C2\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+2sin⁡Asin⁡Bcos⁡(A+B)=sin⁡2(A+B)\sin^2A + \sin^2B + 2\sin A\sin B\cos(A + B) = \sin^2(A + B)

  4. If A+B+C=180∘,A + B + C = 180^\circ, prove that cos⁡2A+cos⁡2B+cos⁡2C+2cos⁡Acos⁡Bcos⁡C=1\cos^2A + \cos^2B + \cos^2C + 2\cos A\cos B\cos C = 1

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

  6. If A+B+C=180∘,A + B + C = 180^\circ, prove that cos⁡2A+cos⁡2B−cos⁡2C=1−2sin⁡Asin⁡Bsin⁡C\cos^2A + \cos^2B - \cos^2C = 1 - 2\sin A\sin B\sin C

  7. If A+B+C=180∘,A + B + C = 180^\circ, prove that cos⁡2A2+cos⁡2B2−cos⁡2C2=2cos⁡A2cos⁡B2sin⁡C2\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∘,A + B + C = 180^\circ, prove that cos⁡2A2+cos⁡2B2+cos⁡2C2=2+2sin⁡A2sin⁡B2sin⁡C2\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=π2,A + B + C = \frac{\pi}{2}, prove that sin⁡2A+sin⁡2B+sin⁡2C=1−2sin⁡Asin⁡Bsin⁡C\sin^2A + \sin^2B + \sin^2C = 1 - 2\sin A\sin B\sin C

  10. If A+B+C=π2,A + B + C = \frac{\pi}{2}, prove that cos⁡2A+cos⁡2B+cos⁡2C=2+2sin⁡Asin⁡Bsin⁡C\cos^2A + \cos^2B + \cos^2C = 2 + 2\sin A\sin B\sin C

  11. If A+B+C=2π,A + B + C = 2\pi, prove that cos⁡2A+cos⁡2B+cos⁡2C−2cos⁡Acos⁡Bcos⁡C=1\cos^2A + \cos^2B + \cos^2C - 2\cos A\cos B\cos C = 1

  12. If A+B=C,A + B = C, prove that cos⁡2A+cos⁡2B+cos⁡2C−2cos⁡Acos⁡Bcos⁡C=1\cos^2A + \cos^2B + \cos^2C - 2\cos A\cos B\cos C = 1

  13. If A+B=π3,A + B = \frac{\pi}{3}, prove that cos⁡2A+cos⁡2B−cos⁡Acos⁡B=34\cos^2A + \cos^2B - \cos A\cos B = \frac{3}{4}

  14. Show that cos⁡2B+cos⁡2(A+B)−2cos⁡Acos⁡Bcos⁡(A+B)\cos^2B + \cos^2(A + B) - 2\cos A\cos B\cos(A + B) is independent of B.B.

  15. If A+B+C=πA + B + C = \pi and A+B=2C,A + B = 2C, prove that 4(sin⁡2A+sin⁡2B−sin⁡Asin⁡B)=34(\sin^2A + \sin^2B - \sin A\sin B) = 3

  16. If A+B+C=2π,A + B + C = 2\pi, prove that cos⁡2B+cos⁡2C−sin⁡2A−2cos⁡Acos⁡Bcos⁡C=0\cos^2B + \cos^2C - \sin^2A - 2\cos A\cos B\cos C = 0

  17. If A+B+C=0,A + B + C = 0, prove that cos⁡2A+cos⁡2B+cos⁡2C=1+2cos⁡Acos⁡Bcos⁡C\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+2cos⁡(B−C)cos⁡(C−A)cos⁡(A−B)\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=π,A + B + C = \pi, prove that sin⁡Acos⁡Bcos⁡C+sin⁡Bcos⁡Ccos⁡A+sin⁡Ccos⁡Acos⁡B=sin⁡Asin⁡Bsin⁡C\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=π,A + B + C = \pi, prove that tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A + \tan B + \tan C = \tan A\tan B\tan C

  21. If A+B+C=π,A + B + C = \pi, prove that tan⁡A2tan⁡B2+tan⁡B2tan⁡C2+tan⁡C2tan⁡A2=1\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=π,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)\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=π,A + B + C = \pi, prove that cot⁡Bcot⁡C+cot⁡Ccot⁡A+cot⁡Acot⁡B=1\cot B\cot C + \cot C\cot A + \cot A\cot B = 1

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

  25. If A,B,C,DA, B, C, D are angles of a quadrilateral, prove that tan⁡A+tan⁡B+tan⁡C+tan⁡Dcot⁡A+cot⁡B+cot⁡C+cot⁡D=tan⁡Atan⁡Btan⁡Ctan⁡D\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=π2,A + B + C = \frac{\pi}{2}, show that cot⁡A+cot⁡B+cot⁡C=cot⁡Acot⁡Bcot⁡C\cot A + \cot B + \cot C = \cot A\cot B\cot C

  27. If A+B+C=π2,A + B + C = \frac{\pi}{2}, show that tan⁡Atan⁡B+tan⁡Btan⁡C+tan⁡Ctan⁡A=1\tan A\tan B + \tan B\tan C + \tan C\tan A = 1

  28. If A+B+C=π,A + B + C = \pi, prove that tan⁡3A+tan⁡3B+tan⁡3C=tan⁡3Atan⁡3Btan⁡3C\tan 3A + \tan 3B + \tan 3C = \tan 3A\tan 3B\tan 3C

  29. If A+B+C=π,A + B + C = \pi, prove that cot⁡A2+cot⁡B2+cot⁡C2=cot⁡A2cot⁡B2cot⁡C2\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=π,A + B + C = \pi, prove that cot⁡A+cot⁡Btan⁡A+tan⁡B+cot⁡B+cot⁡Ctan⁡B+tan⁡C+cot⁡C+cot⁡Atan⁡C+tan⁡A=1\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)\tan(A - B) + \tan(B - C) + \tan(C - A) = \tan(A - B)\tan(B - C)\tan(C - A)

  32. If x+y+z=0,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\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=nπ(nA + B + C = n\pi(n being zero or an integer ),), show that tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A + \tan B + \tan C = \tan A\tan B\tan C

  34. If A+B+C=π,A + B + C = \pi, prove that sin⁡2A+sin⁡2B+sin⁡2C=4sin⁡Asin⁡Bsin⁡C\sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C

  35. If A+B+C=π,A + B + C = \pi, prove that cos⁡A+cos⁡B+cos⁡C−1=4sin⁡A2sin⁡B2sin⁡C2\cos A + \cos B + \cos C - 1 = 4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}

  36. Prove that sin⁡2A+sin⁡2B+sin⁡2Ccos⁡A+cos⁡B+cos⁡C−1=8cos⁡A2cos⁡B2cos⁡C2\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=π,A + B + C = \pi, prove that cos⁡A2+cos⁡B2+cos⁡C2=4cos⁡π−A4cos⁡π−B4cos⁡π−C4\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=π,A + B + C = \pi, prove that sin⁡A2+sin⁡B2+sin⁡C2=1+4sin⁡B+C4sin⁡C+A4sin⁡A+B4\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=π,A + B + C = \pi, prove that sin⁡2A2+sin⁡2B2−sin⁡2C2=1−2cos⁡A2cos⁡B2cos⁡C2\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∘+cos⁡58∘−cos⁡66∘=4cos⁡28∘cos⁡29∘sin⁡33∘1 + \cos 56^\circ + \cos 58^\circ - \cos 66^\circ = 4\cos28^\circ\cos29^\circ\sin 33^\circ

  41. If A+B+C=π,A + B + C = \pi, prove that cos⁡2A+cos⁡2B−cos⁡2C=1−4sin⁡Asin⁡Bcos⁡C\cos 2A + \cos 2B - \cos 2C = 1 - 4\sin A\sin B\cos C

  42. If A+B+C=π,A + B + C = \pi, prove that sin⁡2A+sin⁡2B−sin⁡2C=4cos⁡Acos⁡Bsin⁡C\sin 2A + \sin 2B - \sin 2C = 4\cos A\cos B\sin C

  43. If A+B+C=π,A + B + C = \pi, prove that sin⁡A+sin⁡B+sin⁡C=4cos⁡A2cos⁡B2cos⁡C2\sin A + \sin B + \sin C = 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}

  44. If A+B+C=π,A + B + C = \pi, prove that cos⁡A+cos⁡B−cos⁡C=4cos⁡A2cos⁡B2sin⁡C2−1\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=π,A + B + C = \pi, prove that sin⁡(B+C−A)+sin⁡(C+A−B)+sin⁡(A+B−C)=4sin⁡Asin⁡Bsin⁡C\sin(B + C - A) + \sin(C + A - B) + \sin(A + B - C) = 4\sin A\sin B\sin C

  46. If A+B+C=π,A + B + C = \pi, prove that cos⁡Asin⁡Bsin⁡C+cos⁡Bsin⁡Csin⁡A+cos⁡Csin⁡Asin⁡B=2\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=π,A + B + C = \pi, prove that sin⁡2A+sin⁡2B+sin⁡2Csin⁡A+sin⁡B+sin⁡C=8sin⁡A2 sin⁡B2sin⁡C2\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=π2,x + y + z = \frac{\pi}{2}, prove that cos⁡(x−y−z)+cos⁡(y−z−x)+cos⁡(z−x−y)−4cos⁡xcos⁡ycos⁡z=0\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)+4sin⁡x−y2sin⁡y−z2sin⁡z−x2=0\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∘,A + B + C = 180^\circ, prove that sin⁡(B+2C)+sin⁡(C+2A)+sin⁡(A+2B)=4sin⁡B−C2sin⁡C−A2sin⁡A−B2\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=π,A + B + C = \pi, prove that sin⁡B+C2+sin⁡C+A2+sin⁡A+B2=4cos⁡π−A4cos⁡π−B4cos⁡π−C4\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,xy + yz + zx = 1, prove that x1−x2+y1−y2+z1−z2=4xyz(1−x2)(1−y2)(1−z2)\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,x + y + z = xyz, show that 3x−x31−3x2+3y−y31−3y2+3z−z31−3z2=3x−x31−3x2.3y−y31−3y2.3z−z31−3z2\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,x + y + z = xyz, prove that 2x1−x2+2y1−y2+2z1−z2=2x1−x2.2y1−y2.2z1−z2\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,x + y + z = xyz, prove that x(1−y2)(1−z2)+y(1−z2)(1−x2)+z(1−x2)(1−y2)=4xyzx(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=2π,A + B + C + D = 2\pi, prove that cos⁡A+cos⁡B+cos⁡C+cos⁡D=4cos⁡A+B2cos⁡B+C2cos⁡C+A2\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,A + B + C = 2S, prove that cos⁡2S+cos⁡2(S−A)+cos⁡2(S−B)+cos⁡2(S−C)=2+2cos⁡Acos⁡Bcos⁡C\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=π,A + B + C = \pi, prove that tan⁡2A2+tan⁡2B2+tan⁡2C2≥1\tan^2\frac{A}{2} + \tan^2\frac{B}{2} + \tan^2\frac{C}{2}\geq 1

  59. If A+B+C=π,A + B + C = \pi, prove that (tan⁡A+tan⁡B+tan⁡C)(cot⁡A+cot⁡B+cot⁡C)=1+sec⁡Asec⁡Bsec⁡C(\tan A + \tan B + \tan C)(\cot A + \cot B + \cot C) = 1 + \sec A\sec B\sec C

  60. If A+B+C=π,A + B + C = \pi, prove that (cot⁡B+cot⁡C)(cot⁡C+cot⁡A)(cot⁡A+cot⁡C)=cosec⁡Acosec⁡Bcosec⁡C(\cot B + \cot C)(\cot C + \cot A)(\cot A + \cot C) = \cosec A\cosec B\cosec C

  61. If A+B+C=π,A + B + C = \pi, prove that 12∑sin⁡2A(sin⁡2B+sin⁡2C)=3sin⁡Asin⁡Bsin⁡C\frac{1}{2}\sum \sin^2A(\sin 2B + \sin 2C) = 3\sin A\sin B\sin C

  62. If A+B+C+D=2π,A + B + C + D = 2\pi, prove that cos⁡A−cos⁡B+cos⁡C−cos⁡D=4sin⁡A+B2sin⁡A+D2cos⁡A+C2\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,DA, B, C, D be the angles of a cyclic quadrilateral, prove that cos⁡A+cos⁡B+cos⁡C+cos⁡D=0\cos A + \cos B + \cos C + \cos D = 0

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

  65. If A+B+C=π,A + B + C = \pi, prove that cos⁡A2cos⁡B−C2+cos⁡B2cos⁡C−A2+cos⁡C2cos⁡A−B2=sin⁡A+sin⁡B+sin⁡C\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 △ABC,\triangle ABC, prove that sin⁡3Asin⁡(B−C)+sin⁡3Bsin⁡(C−A)+sin⁡3Csin⁡(A−B)=0\sin 3A\sin(B - C) + \sin 3B\sin(C - A) + \sin3C\sin(A - B) = 0