# 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 $\sin2(\theta + \phi) + \sin2\theta = 2\sin2\phi$