10. Transformation Formulae

10.1. Transformation of products into sums or differences

We know that \(\sin(A + B) = \sin A\cos B + \cos A\sin B\)

and \(\sin(A - B) = \sin A\cos B - \cos A\sin B\)

Adding these, we get \(2\sin A\cos B = \sin(A + B) + \sin(A - B)\)

Subtracting, we get \(2\cos A\sin B = \sin(A + B) - \sin (A - B)\)

We also know that \(\cos (A + B) = \cos A\cos B - \sin A\sin B\)

and \(\cos(A - B) = \cos A\cos B + \sin A\sin B\)

Adding, we get \(2\cos A\cos B = \cos (A + B) + \cos(A - B)\)

Subtrating we get \(2\sin \sin B = \cos (A - B) - \cos(A + B)\)

10.2. Transformation of sums or differences into products

We have \(2\sin A\cos B = \sin(A + B)\sin(A - B)\)

Substituting for \(A + B = C, A - B = D\) so that \(A = \frac{C + D}{2}\) and \(B = \frac{C- D}{2}\)

\(\sin C + \sin D = 2\sin \frac{C + D}{2}\cos \frac{C - D}{2}\)

We also have \(2\cos A\sin B = \sin(A + B) - \sin (A - B)\)

Following similarly \(\sin C - \sin D = 2\cos \frac{C + D}{2}\sin \frac{C - D}{2}\)

For \(2\cos A\cos B = \cos (A + B) + \cos(A - B)\)

we get \(\cos C + \cos D = 2\cos \frac{C + D}{2}\cos \frac{C - D}{2}\)

For \(2\sin \sin B = \cos (A - B) - \cos(A + B)\)

we get \(\cos C - \cos D = 2\sin \frac{C + D}{2}\sin \frac{D - C}{2}\)

10.3. Problems

  1. Find the value of \(\frac{\sin 75^\circ - \sin 15^\circ}{\cos 75^\circ + \cos 15^\circ}\)

  2. Simplify the expression \(\frac{(\cos \theta - \cos 3\theta)(\sin 8\theta + \sin 2\theta)}{(\sin 5\theta - \sin\theta)(\cos 4\theta - \cos 6\theta)}\)

Prove that

  1. \(\frac{\sin7\theta - \sin5\theta}{\cos7\theta + \cos5\theta} = \tan\theta\)

  2. \(\frac{\cos6\theta - \cos4\theta}{\sin6\theta + \sin4\theta} = -\tan\theta\)

  3. \(\frac{\sin A + \sin 3A}{\cos A + \cos 3A} = \tan 2A\)

  4. \(\frac{\sin 7A - \sin A}{\sin 8A - \sin 2A} = \cos 4A\sec 5A\)

  5. \(\frac{\cos 2B + \cos 2A}{\cos 2B - \cos 2A} = \cot(A + B)\cot(A - B)\)

  6. \(\frac{\sin 2A + \sin 2B}{\sin 2A - \sin 2B} = \frac{\tan(A + B)}{\tan(A - B)}\)

  7. \(\frac{\sin A + \sin 2A}{\cos A - \cos 2A} = \cot \frac{A}{2}\)

  8. \(\frac{\sin 5A - \sin 3A}{\cos 3A + \cos 5A} = \tan A\)

  9. \(\frac{\cos 2B - \cos 2A}{\sin 2B + \sin 2A} = \tan(A - B)\)

  10. \(\cos (A + B) + \sin(A - B) = 2\sin(45^\circ + A)\cos(45^\circ + B)\)

  11. \(\frac{\cos 3A - \cos A}{\sin 3A - \sin A} + \frac{\cos 2A - \cos 4A}{\sin 4A - \sin 2A} = \frac{\sin A}{\cos 2A\cos 3A}\)

  12. \(\frac{\sin (4A - 2B) + \sin (4B - 2A)}{\cos (4A - 2B) + \cos (4B - 2A)} = \tan(A + B)\)

  13. \(\frac{\tan 5\theta + \tan 3\theta}{\tan 5\theta - \tan 3\theta} = 4\cos 2\theta\cos 4\theta\)

  14. \(\frac{\cos 3\theta + 2\cos5\theta + \cos 7\theta}{\cos\theta + 2\cos3\theta + \cos 5\theta} = \cos 2\theta - \sin 2\theta\tan 3\theta\)

  15. \(\frac{\sin A + \sin 3A + \sin 5A + \sin 7A}{\cos A + \cos 3A + \cos 5A + \cos 7A} = \tan 4A\)

  16. \(\frac{\sin (\theta + \phi) - 2\sin\theta + \sin (\theta - \phi)}{\cos (\theta + \phi) - 2\cos \theta + \cos(\theta - \phi)} = \tan\theta\)

  17. \(\frac{\sin A + 2\sin 3A + \sin 5A}{\sin 3A + 2\sin 5A + \sin 7A} = \frac{\sin 3A}{\sin 5A}\)

  18. \(\frac{\sin(A - C) + 2\sin A + \sin(A + C)}{\sin (B - C) + 2\sin B + \sin(B + C)} = \frac{\sin A}{\sin B}\)

  19. \(\frac{\sin A - \sin 5A + \sin 9A - \sin 13A}{\cos A - \cos 5A - \cos 9A + \cos 13 A} = \cot 4A\)

  20. \(\frac{\sin A + \sin B}{\sin A - \sin B} = \tan \frac{A + B}{2}\cot \frac{A - B}{2}\)

  21. \(\frac{\cos A + \cos B}{\cos B - \cos A} = \cot \frac{A + B}{2}\cot \frac{A - B}{2}\)

  22. \(\frac{\sin A + \sin B}{\cos A + \cos B} = \tan \frac{A + B}{2}\)

  23. \(\frac{\sin A - \sin B}{\cos B - \cos A} = \cot \frac{A + B}{2}\)

  24. \(\frac{\cos(A + B + C) + \cos(-A + B + C) + \cos(A - B + C) + \cos(A + B - C)}{\sin(A + B + C)+\sin(-A + B + C) - \sin(A - B + C) + \sin(A + B - C)} = \cot B\)

  25. \(\cos 3A + \cos 5A + \cos 7A + \cos 15A = 4 \cos 4A\cos 5A \cos 6A\)

  26. \(\cos(-A + B + C) + \cos(A - B + C) + \cos(A + B - C) + \cos(A + B + C) = 4\cos A\cos B\cos C\)

  27. \(\sin 50^\circ - \sin 70^\circ + \sin 10^\circ = 0\)

  28. \(\sin 10^\circ + \sin 20^\circ + \sin 40^\circ + \sin 50^\circ = \sin 70^\circ + \sin 80^\circ\)

  29. \(\sin\alpha + \sin 2\alpha + \sin 4\alpha + \sin 5\alpha = 4\cos \frac{\alpha}{2}\cos \frac{3\alpha}{2}\sin 3\alpha\)

Simplify:

  1. \(\cos\left[\theta + \left(n - \frac{3}{2}\right)\phi\right] - \cos\left[\theta + \left(n + \frac{3}{2}\right)\phi\right]\)

  2. \(\sin\left[\theta + \left(n - \frac{3}{2}\right)\phi\right] + \sin\left[\theta + \left(n + \frac{3}{2}\right)\phi\right]\)

Express as a sum or difference the following:

  1. \(2\sin5\theta\sin7\theta\)

  2. \(2\cos7\theta\sin5\theta\)

  3. \(2\cos 11\theta\cos 3\theta\)

  4. \(2\sin54^\circ\sin66^\circ\)

Prove that

  1. \(\sin\frac{\theta}{2}\sin\frac{7\theta}{2} + \sin \frac{3\theta}{2}\sin\frac{11\theta}{2} =\sin 2\theta\sin 5\theta\)

  2. \(\cos 2\theta\cos \frac{\theta}{2} -\cos3\theta\cos\frac{9\theta}{2} = \sin5\theta\sin\frac{5\theta}{2}\)

  3. \(\sin A\sin(A + 2B) - \sin B\sin(B + 2A) = \sin(A - B)\sin(A + B)\)

  4. \((\sin 3A + \sin A)\sin A + (\cos 3A - \cos A)\cos A = 0\)

  5. \(\frac{2\sin(A - C)\cos C - \sin(A - 2C)}{2\sin(B - C)\cos C - \sin(B - 2C)} = \frac{\sin A}{\sin B}\)

  6. \(\frac{\sin A\sin 2A + \sin 3A\sin 6A + \sin4A\sin 13A}{\sin A\cos2A + \sin 3A\cos 6A + \sin 4A\cos 13A} = \tan 9A\)

  7. \(\frac{\cos 2A\cos 3A - \cos 2A\cos 7A + \cos A\cos 10A}{\sin 4A\sin 3A - \sin 2A\sin 5A + \sin 4A\sin 7A} =\cot 6A\cot 5A\)

  8. \(\cos(36^\circ - A)\cos(36^\circ + A) + \cos(54^\circ + A)\cos(54^\circ - A) = \cos 2A\)

  9. \(\cos A\sin(B - C) + \cos B\sin(C - A) + \cos C\sin(A - B) = 0\)

  10. \(\sin(45^\circ + A)\sin(45^\circ - A) = \frac{1}{2}\cos 2A\)

  11. \(\sin(\beta - \gamma)\cos(\alpha - \delta) + \sin(\gamma - \alpha)\cos(\beta - \delta) + \sin(\alpha - \beta)\cos(\gamma - \delta) = 0\)

  12. \(2\cos\frac{\pi}{13}\cos \frac{9\pi}{13} + \cos \frac{3\pi}{13} + \cos \frac{5\pi}{13} = 0\)

  13. \(\cos 55^\circ + \cos65^\circ + \cos 175^\circ = 0\)

  14. \(\cos 18^\circ -\sin 18^\circ = \sqrt{2}\sin 27^\circ\)

  15. \(\frac{\sin A + \sin 2A + \sin 4A + \sin 5A}{\cos A + \cos 2A + \cos 4A + \cos 5A} = \tan 3A\)

  16. \(\left(\frac{\cos A + \cos B}{\sin A - \sin A}\right)^n + \left(\frac{\sin A + \sin B}{\cos A - \cos B}\right)^n = 2\cot^n \frac{A - B}{2}\) or \(0\) accordingh as \(n\) is even or odd.

  17. If \(\alpha, \beta, \gamma\) are in A.P., show that \(\cos\beta = \frac{\sin\alpha - \sin\gamma}{\cos\gamma - \cos\alpha}\)

  18. If \(\sin\theta + \sin\phi = \sqrt{3}(\cos\phi - \cos\theta)\) prove that \(\sin3\theta + \sin3\phi = 0\)

  19. \(\sin 65^\circ + cos 65^\circ = \sqrt{2}\cos 20^\circ\)

  20. \(\sin 47^\circ + \cos 77^\circ = \cos 17^\circ\)

  21. \(\frac{\cos 10^\circ - \sin 10^\circ}{\cos 10^\circ + \sin 10^\circ} = \tan 35^\circ\)

  22. \(\cos 80^\circ + \cos 40^\circ - cos 20^\circ = 0\)

  23. \(\cos\frac{\pi}{5} + \cos \frac{2\pi}{5} + \cos\frac{6\pi}{5} + \cos \frac{7\pi}{5} = 0\)

  24. \(\cos\alpha + \cos\beta + \cos\gamma + \cos(\alpha + \beta + \gamma) = 4\cos\frac{\alpha + \beta}{2}\cos\frac{\beta + \gamma}{2}\cos \frac{\gamma + \alpha}{2}\)

  25. If \(\sin\alpha - \sin\beta = \frac{1}{3}\) and \(\cos\beta - \cos\alpha = \frac{1}{2},\) prove that \(\cot\frac{\alpha + \beta}{2} = \frac{2}{3}\)

  26. If \(\cosec A + sec A = \cosec B + \sec B,\) prove that \(\tan A\tan B = \cot \frac{A + B}{2}\)

  27. If \(\sec(\theta + \alpha) + \sec(\theta - \alpha) = 2\sec\theta,\) show that \(\cos^2\theta = 1 + \cos\alpha\)

  28. Show that \(\sin50^\circ\cos85^\circ = \frac{1 - \sqrt{2}\sin 35^\circ}{2\sqrt{2}}\)

  29. Prove that \(\sin 20^\circ \sin 40^\circ\sin 80^\circ = \frac{\sqrt{3}}{8}\)

  30. Prove that \(\sin A\sin(60^\circ - A)\sin(60^\circ + A) = \frac{1}{4}\sin 3A\)

  31. If \(\alpha + \beta = 90^\circ,\) find the maximum value of \(\sin\alpha\sin\beta\)

  32. Prove that \(\sin 25^\circ\cos 115^\circ = \frac{1}{2}(\sin 40^\circ - 1)\)

  33. Prove that \(\sin 20^\circ \sin 40^\circ\sin 60^\circ \sin80^\circ = \frac{3}{16}\)

  34. Prove that \(\cos 20^\circ\cos40^\circ\cos80^\circ = \frac{1}{8}\)

  35. Prove that \(\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ = 3\)

  36. Prove that \(\cos10^\circ\cos30^\circ\cos50^\circ\cos70^\circ = \frac{3}{16}\)

  37. Prove that \(4\cos\theta\cos\left(\frac{\pi}{3} + \theta\right)\cos\left(\frac{\pi}{3} - \theta\right) = \cos3\theta\)

  38. Prove that \(\tan\theta\tan(60^\circ - \theta)\tan(60^\circ + \theta) = \tan3\theta\)

  39. If \(\alpha + \beta = 90^\circ,\) show that the maximum value of \(\cos\alpha\cos\beta\) is \(\frac{1}{2}\)

  40. If \(\cos\alpha = \frac{1}{\sqrt{2}}, \sin\beta = \frac{1}{\sqrt{3}},\) show that \(\tan\frac{\alpha + \beta}{2}\cot\frac{\alpha - \beta}{2} = 5 + 2\sqrt{6}\) or \(5- 2\sqrt{6 }\)

  41. If \(x\cos\theta = y\cos\left(\theta + \frac{2\pi}{3}\right) = z\cos\left(\theta + \frac{4\pi}{3}\right),\) prove that \(xy + yz + xz = 0\)

  42. If \(\sin\theta = n\sin(\theta + 2\alpha),\) prove that \(\tan(\theta + \alpha) = \frac{1 + n}{1 - n}\tan\alpha\)

  43. If \(\frac{\sin(\theta + \alpha)}{\cos(\theta - \alpha)} = \frac{1 - m}{1 + m},\) prove that \(\tan\left(\frac{\pi}{4} - \theta\right)\tan\left(\frac{\pi}{4} - \alpha\right) = m\)

  44. If \(y\sin\phi = x\sin(2\theta + \phi),\) show that \((x + y)\cot(\theta + \phi) = (y - x)\cot\theta\)

  45. If \(\cos(\alpha + \beta)\sin(\gamma + \delta) = \cos(\alpha - beta)\sin(\gamma - \delta),\) prove that \(\cot\alpha\cot\beta\cot\gamma = cot\delta\)

  46. If \(\frac{\cos(A - B)}{\cos(A + B)} + \frac{\cos(C + D)}{\cos(C - D)} = 0,\) prove that \(\tan A\tan B\tan C\tan D = -1\)

  47. If \(\tan(\theta + \phi) = 3\tan\theta,\) prove that \(\sin(2\theta + \phi) = 2\sin\phi\)

  48. If \(\tan(\theta + \phi) = 3\tan\theta,\) prove that \(\sin2(\theta + \phi) + \sin2\theta = 2\sin2\phi\)