# 12. Multiple and Submultiple Angles¶

## 12.1. Multiple Angles¶

An angle of the form $nA,$ where $n$ is an integer is called a multiple angle. For example, $2A, 3A, 4A, \ldots$ are multiple angles of $A.$

### 12.1.1. Trigonometrical Ratios of $2A$¶

From previous chapter we know that $\sin(A + B) = \sin A\cos B + \cos A\sin B$

Substituting $B = A,$ we get

$\sin 2A = 2\sin A\cos A$

Similarly, $\cos 2A = \cos^2A - \sin^2A = 2\cos^2A - 1 = 1 - 2\sin^2A$ (recall formula from previous chapter and substitute $B = A$ $\cos^2A = 1 -\sin^2A$ and $\sin^2A = 1 - \cos^2a$)

Also, $\tan 2A = \frac{2\tan A}{1 - \tan^2A}$ (recall formula from previous chapter and put $B = A$)

### 12.1.2. $\sin 2A$ and $\cos 2A$ in terms of $\tan A$¶

$\sin 2A = \frac{2\sin A\cos A}{\sin^2A + \cos^2A}[\because \sin^2A + \cos^2A = 1]$

Dividing both numerator and denominator by $\cos^2A,$ we get

$\sin 2A = \frac{2\tan A}{1 + \tan^2A}$

$\cos A = \cos^2A - \sin^2A = \frac{\cos^2A - \sin^2A}{\cos^2A + \sin^2A}[\because \sin^2A + \cos^2A = 1]$

Dividing both numerator and denominator by $\cos^2A,$ we get

$\cos 2A = \frac{1 - \tan^2A}{1 + \tan^2A} = \frac{\cot^2A - 1}{\cot^2A + 1}$

### 12.1.3. Trigonometrical Ratios of $3A$¶

$\sin 3A = \sin2A\cos A + \cos 2A\sin A = 2\sin A\cos^2 A + \cos^2A\sin A - \sin^3A$

$= 2\sin A(1 - \sin^2A) + (1 - 2\sin^2A)\sin A - \sin^3A$

$= 3\sin A - 4\sin^3A$

$\cos 3A = \cos2A\cos A - \sin 2A\sin A = (2\cos^2A - 1)\cos A - 2\sin^2 A\cos A$

$= 2\cos^3A - \cos A - 2(1 - \cos^2A)\cos A$

$= 4\cos^3 A - 3\cos A$

We know that $\tan(A + B + C) = \frac{\tan A + \tan B + \tan C - \tan A\tan B\tan C}{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}$

Putting $B = A$ and $C = A,$ we get

$\tan 3A = \frac{3\tan A - \tan^3A}{1 - 3\tan^2A}$

Similarly $\cot 3A = \frac{\cot^3 A - 3\cot A}{3\cot^2A - 1}$

### 12.1.4. Some Important Formulae¶

1. $\cos2A = 1 - 2\sin^2A \Rightarrow \sin^2A = \frac{1}{2}(1 - \cos2A)$

2. $\cos2A = 2\cos^2 - 1 \Rightarrow \cos^2A = \frac{1}{2}(1 + \cos2A)$

3. $\sin 3A = 3\sin A - 4\sin^3A \Rightarrow sin^3A = \frac{1}{2}(3\sin A - \sin3A)$

4. $\cos 3A = 4\cos^3A - 3\cos A \Rightarrow \cos^3A = \frac{1}{4}(\cos3A + 3\cos A)$

## 12.2. Submultiple Angles¶

An angle of the form $\frac{A}{n},$ where $n$ is an integer is called a submultiple angle. For exmaple, $\frac{A}{2}, \frac{A}{3}, \frac{A}{4}, \ldots$ are submultiple angles of $A.$

### 12.2.1. Trigonometrical Ratios of $A/2$¶

We know that, $\sin 2A = 2\sin A\cos A.$ Putting $A=A/2,$ we get

$\sin A = 2\sin A/2\cos A/2$

$\cos 2A = \cos^2A - \sin^A.$ Putting $A = A/2,$ we get

$\cos A - \cos^2\frac{A}{2} - \sin^2\frac{A}{2}$

$\cos 2A = 2\cos^2A - 1.$ Putting $A = A/2,$ we get

$\cos A = 2\cos^2\frac{A}{2} - 1$

$\cos 2A = 1 - 2\sin^2A.$ Putting $A = A/2,$ we get

$\cos A = 1 - 2\sin^2\frac{A}{2}$

$\tan 2A = \frac{2\tan A}{1 - \tan^2A}.$ Putting $A = A/2,$ we get

$\tan A = \frac{2\tan \frac{A}{2}}{1 - \tan^2\frac{A}{2}}$

$\sin 2A = \frac{2\tan A}{1 + \tan^2A} \therefore \sin A = \frac{2\tan \frac{A}{2}}{1 + \tan^2\frac{A}{2}}$

$\cos 2A = \frac{1 - \tan^2A}{1 + \tan^2A} \therefore \cos A = \frac{1 - \tan^2\frac{A}{2}}{1 + \tan^2\frac{A}{2}}$

$\cot 2A = \frac{\cot^2A - 1}{2\cot A} \therefore \cot A = \frac{\cot^2\frac{A}{2} - 1}{2\cot \frac{A}{2}}$

### 12.2.2. Trigonometrical Ratios of $A/3$¶

$\sin 3A = 3\sin A - 4\sin^3A.$ Putting $A = \frac{A}{3},$ we get

$\sin A = \frac{3}{\sin \frac{A}{3}} - 4\sin^3\frac{A}{3}$

$\cos 3A = 4\cos^3A - 3\cos A$. Putting $A = \frac{A}{3},$ we get

$\cos A = 4\cos^3\frac{A}{3} - 3\cos \frac{A}{3}$

$\tan 3A = \frac{3\tan A - \tan^3A}{1 - 3\tan^2A}$

$\tan A = \frac{3\tan\frac{A}{3} - \tan^3\frac{A}{3}}{1 - 3\tan^2\frac{A}{3}}$

### 12.2.3. Values of $cos A/2, \sin A/2$ and $\tan A/2$ in terms of $\cos A$¶

$\cos^2\frac{A}{2} = \frac{1 + \cos A}{2} \therefore \cos \frac{A}{2} = \sqrt{\frac{1 + \cos A}{2}}$

$\sin^2\frac{A}{2} = \frac{1 - \cos A}{}2 \therefore \sin \frac{A}{2} = \sqrt{\frac{1 - \cos A}{2}}$

$\tan^2\frac{A}{2} = \frac{1 - \cos A}{1 + \cos A} \therefore \tan\frac{A}{2} = \sqrt{\frac{1 - \cos A}{1 + \cos A}}$

### 12.2.4. Values of $\sin A/2$ and $\cos A/2$ in terms of $\sin A$¶

$\left(\cos \frac{A}{2} + \sin\frac{A}{2}\right)^2 = \cos^2\frac{A}{2} + \sin^2\frac{A}{2} + 2\cos\frac{A}{2}\sin\frac{A}{2}$

$= 1 + \sin A \Rightarrow \cos \frac{A}{2} + \sin \frac{A}{2} = \sqrt{1 + \sin A}$

Similarly, $\cos \frac{A}{2} - \sin \frac{A}{2} = \sqrt{1 - \sin A}$

Adding, we get $\cos \frac{A}{2} = \pm\frac{1}{2}\sqrt{1 + \sin A} \pm\frac{1}{2}\sqrt{1 - \sin A}$

Subtracting, we get $\cos \frac{A}{2} = \pm\frac{1}{2}\sqrt{1 + \sin A} \mp\frac{1}{2}\sqrt{1 - \sin A}$

### 12.2.5. Value of $\sin 18^\circ$ and $\cos 72^\circ$¶

Let $A = 18^\circ,$ then $\sin 5A = 90^\circ \therefore 2A + 3A = 90^\circ$

$\sin2A = \sin(90^\circ - \sin 3A) \therefore 2\sin A\cos A = 4\cos^3A - 3\cos A$

Dividing both sides by $\cos A,$ we get

$2\sin A = 4\cos^2A - 3 = 4(1 - \sin^2A) - 3$

$4\sin^2A + 2\sin A - 1 = 0$

$\sin A = \frac{-1\pm\sqrt{5}}{4}$

However, since $A= 18^\circ\therefore \sin A > 0$

$\therefore \sin18^\circ = \frac{-1 + \sqrt{5}}{4}$

$\therefore \sin(90^\circ - 18^\circ) = \cos72^\circ = \frac{\sqrt{5} - 1}{4}$

### 12.2.6. Value of $\cos 18^\circ$ and $\sin 72^\circ$¶

$\cos^218^\circ = 1 - \sin^218^\circ = 1 - \left(\frac{\sqrt{5} - 1}{4}\right)^2$

$= \frac{10 + 2\sqrt{5}}{16}\therefore \cos18^\circ = \frac{1}{4}\sqrt{10 + 2\sqrt{5}}[\because \cos18^\circ > 0]$

$\cos(90^\circ - 18^\circ) = \sin72^\circ = \frac{1}{4}\sqrt{10 + 2\sqrt{5}}$

### 12.2.7. Value of $\tan 18^\circ$ and $\tan 72^\circ$¶

$\tan 18^\circ = \frac{\sin18^\circ}{\cos18^\circ} = \frac{\sqrt{5} - 1}{\sqrt{10 + 2\sqrt{5}}}$

$\tan18^\circ\cot18^\circ = 1\Rightarrow \tan72^\circ = \frac{1}{\tan18^\circ} = \frac{\sqrt{10 + 2\sqrt{5}}}{\sqrt{5} - 1}$ x Value of $\cos 36^\circ$ and $\sin 54^\circ$ ——————————————————– $\cos 36^\circ = 1 - 2\sin^218^\circ = 1 - 2\left(\frac{\sqrt{5} - 1}{4}\right)^2$

$= \frac{\sqrt{5} + 1}{4}$

$\sin 54^\circ = \sin(90^\circ - 36^\circ) = \cos36^\circ = \frac{\sqrt{5} + 1}{4}$

### 12.2.8. Value of $\sin 36^\circ$ and $\cos 54^\circ$¶

$\sin36^\circ = 1 - \cos^236^\circ = 1 - \left(\frac{\sqrt{5} + 1}{4}\right)^2$

$= \frac{1}{4}\sqrt{10 - 2\sqrt{5}}$

$\cos54^\circ = \cos(90^\circ - 36^\circ) = \sin36^\circ = \frac{1}{4}\sqrt{10 - 2\sqrt{5}}$

Several other angles like, $9^\circ, 15^\circ, 22\frac{1}{2}^\circ, 7\frac{1}{2}^\circ$ etc can be found similarrly.

## 12.3. Problems¶

1. Find the value of $\sin 2A,$ when

1. $\cos A = \frac{3}{5}$

2. $\sin A = \frac{12}{13}$

3. $\tan A = \frac{16}{63}$

2. Find the value of $\cos 2A,$ when

1. $\cos A = \frac{15}{17}$

2. $\sin A = \frac{4}{5}$

3. $\tan A = \frac{5}{12}$

3. If $\tan A = \frac{b}{a},$ find the value of $a\cos 2A+ b\sin 2A$

Prove that

1. $\frac{\sin 2A}{1 + \cos 2A} = \tan A$

2. $\frac{\sin 2A}{1 - \cos 2A} = \cot A$

3. $\frac{1 - \cos 2A}{1 + \cos 2A} = \tan^2A$

4. $\tan A + \cot A = 2\cosec 2A$

5. $\tan A - \cot A = -2\cot2A$

6. $\cosec 2A + \cot 2A = \cot A$

7. $\frac{1 - \cos A + \cos B - \cos(A + B)}{1 + \cos A - \cos B - \cos(A + B)} = \tan\frac{A}{2}\cot\frac{B}{2}$

8. $\frac{\cos A}{1 \mp \sin A} = \tan\left(45^\circ \pm \frac{A}{2}\right)$

9. $\frac{\sec 8A - 1}{\sec 4A - 1} = \frac{\tan 8A}{\tan 2A}$

10. $\frac{1 + \tan^2(45^\circ - A)}{1 - \tan^2(45^\circ - A)} = \cosec 2A$

11. $\frac{\sin A + \sin B}{\sin A - \sin B} = \frac{\tan \frac{A + B}{2}}{\tan \frac{A - B}{2}}$

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

13. $\tan\left(\frac{\pi}{4} + A\right) - \tan\left(\frac{\pi}{4} - A\right) = 2\tan 2A$

14. $\frac{\cos A + \sin A}{\cos A - \sin A} - \frac{\cos A - \sin A}{\cos A + \sin A} = 2\tan 2A$

15. $\cot (A + 15^\circ) - \tan(A - 15^\circ) = \frac{4\cos 2A}{1 + 2\sin 2A}$

16. $\frac{\sin A + \sin2A}{1 + \cos A + \cos 2A} = \tan A$

17. $\frac{1 + \sin A - \cos A }{1 + \sin A + cos A} = \tan \frac{A}{2}$

18. $\frac{\sin(n + 1)A - \sin(n - 1)A}{\cos(n + 1)A + 2\cos nA + \cos(n - 1)A} = \tan \frac{A}{2}$

19. $\frac{\sin(n + 1)A + 2\sin nA + \sin(n - 1)A}{\cos(n - 1) - \cos(n + 1)A} = \cot \frac{A}{2}$

20. $\sin(2n + 1)A\sin A = \sin^2(n + 1)A - \sin^2nA$

21. $\frac{\sin(A + 3B) + \sin(3A + B)}{\sin 2A + \sin 2B} = 2\cos(A + B)$

22. $\sin 3A + \sin 2A - \sin A = 4\sin A\cos \frac{A}{2}\cos \frac{3A}{2}$

23. $\tan 2A = (\sec 2A + 1)\sqrt{\sec^2A - 1}$

24. $\cos^32A + 3\cos 2A = 4(\cos^6A - \sin^6A)$

25. $1 + \cos^22A = 2(\cos^4A + \sin^4A)$

26. $\sec^2A(1 + \sec2A) = 2\sec2A$

27. $\cosec A - 2\cot 2A\cos A = 2\sin A$

28. $\cot A = \frac{1}{2}\left(\cot\frac{A}{2} - \tan\frac{A}{2}\right)$

29. $\sin A\sin(60^\circ - A)\sin(60^\circ + A) = \frac{1}{4}\sin 3A$

30. $\cos A\cos(60^\circ - A)\cos(60^\circ + A) = \frac{1}{4}\cos 3A$

31. $\cot A + \cot(60^\circ + A) - \cot(60^\circ - A) = 3\cot 3A$

32. $\cos 4A = 1 - 8\cos^2A + 8\cos^4A$

33. $\sin 4A = 4\sin A\cos^3A - 4\cos A\sin^3A$

34. $\cos 6A = 32\cos^6A - 48\cos^4A + 18\cos^2A - 1$

35. $\tan 3A\tan 2A\tan A = \tan 3A - \tan 2A - \tan A$

36. $\frac{2\cos2^nA + 1}{2\cos A + 1} = (2\cos A - 1)(2\cos 2A - 1)(2\cos2^2A - 1)\ldots(2\cos2^{n - 1} - 1)$

37. If $\tan A= \frac{1}{7}, \sin B = \frac{1}{\sqrt{10}},$ prove that $A + 2B = \frac{\pi}{4},$ where $0 < A < \frac{\pi}{4}$ and $0 < B < \frac{\pi}{4}$

Prove that

1. $\tan\left(\frac{\pi}{4} + A\right) + \tan\left(\frac{\pi}{4} - A\right) = 2\sec2A$

2. $\sqrt{3}\cosec 20^\circ - \sec 20^\circ = 4$

3. $\tan A + 2\tan 2A + 4\tan 4A + 8\cot 8A = \cot A$

4. $\cos^2A + \cos^2\left(\frac{2\pi}{3} - A\right) + \cos^2\left(\frac{2\pi}{3} + A\right) = \frac{3}{2}$

5. $2\sin^2A + 4\cos (A + B)\sin A\sin B + \cos2(A + B)$ is idnependent of $A.$

6. If $\cos A = \frac{1}{2}\left(a + \frac{1}{a}\right),$ show that $\cos 2A = \frac{1}{2}\left(a^2 + \frac{1}{a^2}\right)$

Prove that

7. $\cos^2A + \sin^2A\cos 2B = \cos^2B + \sin^2B\cos 2A$

8. $1 + \tan A\tan 2A = \sec 2A$

9. $\frac{1 + \sin 2A}{1 - \sin 2A} = \left(\frac{1 + \tan A}{1 - \tan A}\right)^2$

10. $\frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} = 4$

11. $\cot^2A - \tan^2A = 4\cot2A\cosec 2A$

12. $\frac{1 +\sin 2A}{\cos2A} = \frac{\cos A + \sin A}{\cos A - \sin A} = \tan\left(\frac{\pi}{4} + A\right)$

13. $\cos^6A - \sin^6A = \cos2A\left(1 - \frac{1}{4}\sin^22A\right)$

14. $\cos^2A + \cos^2\left(\frac{\pi}{3} + A\right) + \cos^2\left(\frac{\pi}{3} - A\right)= \frac{3}{2}$

15. $(1 + \sec2A)(1 + \sec2^2A)(1 + \sec2^3A) \ldots (1 + \sec2^nA) = \frac{\tan2^nA}{\tan A}$

16. $\frac{\sin2^nA}{\sin A} = 2^n\cos A\cos 2A\cos 2^2A\ldots\cos2^{n - 1}A$

17. $3(\sin A - \cos A)^4 + 6(\sin A + \cos A)^2 + 4(\sin^6A + \cos^6A) = 13$

18. $2(\sin^6A + \cos^6A) - 3(\sin^4A + \cos^4A) + 1 = 0$

19. $\cos^2A + \cos^2(A + B) -2\cos A\cos B\cos(A + B)$ if independent of $A.$

20. $\cos^3A\cos 3A + \sin^3A\sin 3A = \cos^32A$

21. $\tan A\tan(60^\circ - A)\tan(60^\circ + A) = \tan 3A$

22. $\sin^2A + \sin^3\left(\frac{2\pi}{3} + A\right) + \sin^3\left(\frac{4\pi}{3} + A\right) = -\frac{3}{4}\sin 3A$

23. $4(\cos^310^\circ + \sin^320^\circ) = 3(\cos 10\circ + \sin 20^\circ)$

24. $\sin A\cos^3A - \cos A\sin^3A = \frac{1}{4}\sin 4A$

25. $\cos^3A\sin3A + \sin^3A\cos 3A = \frac{3}{4}\sin 4A$

26. $\sin A\sin(60^\circ + A)\sin(A + 120^\circ) = \sin 3A$

27. $\cot A + \cot(60^\circ + A) + \cot(120^\circ + A) = 3\cot 3A$

28. $\cos 5A = 16\cos^5A - 20\cos^3A + 5\cos A$

29. $\sin 5A = 5\sin A - 20\sin^3A + 16\sin^5A$

30. $\cos 4A - \cos 4B = 8(\cos A - \cos B)(\cos A + \cos B)(\cos A - \sin B)(\cos A + \sin B)$

31. $\tan 4A = \frac{4\tan A - 4\tan^3A}{1 - 6\tan^2A + \tan^4A}$

32. If $2\tan A = 3\tan B,$ prove that $\tan (A- B) = \frac{\sin 2B}{5 - \cos 2B}$

33. If $\sin A + \sin B = x$ and $\cos A + \cos B = y,$ show that $\sin(A + B) = \frac{2xy}{x^2 + y^2}$

34. If $A= \frac{\pi}{2^n + 1},$ prove that $\cos A.\cos 2A. \cos2^2A.\ldots.\cos2^{n - 1}A = \frac{1}{2^n}$

35. If $\tan A = \frac{y}{x},$ prove that $x\cos 2A + y\sin 2A = x$

36. If $\tan^2A = 1 + 2\tan^2B,$ prove that $\cos 2B = 1 + 2\cos 2A$

37. If $A$ and $B$ lie between $0$ and $\frac{\pi}{2}$ and $\cos 2A = \frac{3\cos 2B - 1}{3 - \cos 2B},$ prove that $\tan A = \sqrt{2}\tan B$

38. If $\tan B = 3\tan A,$ prove that $\tan(A + B) = \frac{2\sin 2B}{1 + \cos 2B}$

39. If $x\sin A = y\cos A,$ prove that $\frac{x}{\sec 2A} + \frac{y}{\cosec 2A} = x$

40. If $\tan A = \sec 2B,$ prove that $\sin 2A = \frac{1 - \tan^4B}{1 + \tan^4B}$

41. If $A = \frac{\pi}{3},$ prove that $\cos A.\cos 2A. \cos 3A.\cos 4A.\cos 5A.\cos 6A = -\frac{1}{16}$

42. If $A = \frac{\pi}{15},$ prove that $\cos2A.\cos4A.\cos8A.\cos14A = \frac{1}{16}$

43. If $\tan A\tan B = \sqrt{\frac{a - b}{a + b}},$ prove that $(a - b\cos2A)(a - b\cos2B) = a^2 - b^2$

44. If $\sin A = \frac{1}{2}$ and $\sin B = \frac{1}{3},$ find the value of $\sin(A + B)$ and $\sin(2A + 2B)$

45. If $\cos A = \frac{11}{61}$ and $\sin B = \frac{4}{5},$ find the value of $\sin^2 \frac{A - B}{2}$ and $cos^2\frac{A + B}{2},$ the angle of $A$ and $B$ being positive acute angles.

46. Given $\sec A = \frac{5}{4},$ find $\tan\frac{A}{2}$ and $\tan A.$

47. If $\cos A = .3,$ find the value of $\tan \frac{A}{2},$ and explain the resulting ambiguity.

48. If $\sin A + \sin B = x$ and $\cos A + \cos B = y,$ find the value of $\tan \frac{A - B}{2}$

Prove that

1. $(\cos A + \cos B)^2 + (\sin A - \sin B)^2 = 4\cos^2 \frac{A + B}{2}$

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

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

4. $\sin^2\left(\frac{\pi}{8} + \frac{A}{2}\right) - \sin^2\left(\frac{\pi}{8} -\frac{A}{2}\right) = \frac{1}{\sqrt{2}}\sin A$

5. $(\tan 4A + \tan 2A)(1 - \tan^23A\tan^2A) = 2\tan 3A\sec^2A$

6. $\left(1 + \tan \frac{A}{2} - \sec\frac{A}{2}\right)\left(1 + \tan \frac{A}{2} + \sec\frac{A}{2}\right) = \sin A\sec^2\frac{A}{2}$

7. $\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \frac{A}{2}$

8. $\frac{1 - \tan \frac{A}{2}}{1 + \tan \frac{A}{2}} = \frac{1 + \sin A}{\cos A} = \tan \left(\frac{\pi}{4} + \frac{A}{2}\right)$

9. $\cos^4\frac{\pi}{8} + \cos^4 \frac{3\pi}{8} + \cos^4\frac{5\pi}{8} + \cos^4\frac{7\pi}{8}= \frac{3}{2}$

10. $\frac{2\sin A - \sin2A}{2\sin A + \sin 2A} = \tan^2\frac{A}{2}$

11. $\cot \frac{A}{2} - \tan \frac{A}{2} = 2\cot A$

12. $\frac{1 + \sin A}{1 - \sin A} = \tan^2\left(\frac{\pi}{4} + \frac{A}{2}\right)$

13. $\sec A + \tan A = \tan\left(\frac{\pi}{4} + \frac{A}{2}\right)$

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

15. $\tan \left(\frac{\pi}{4} - \frac{A}{2}\right) = \sec A - \tan A = \sqrt{\frac{1 - \sin A}{1 + \sin A}}$

16. $\cosec\left(\frac{\pi}{4} + \frac{A}{2}\right)\cosec \left(\frac{\pi}{4} - \frac{A}{2}\right) = 2\sec A$

17. $\cos^2\frac{\pi}{8} + \cos^2\frac{3\pi}{8} + \cos^2\frac{5\pi}{8} + \cos^2\frac{7\pi}{8} = 2$

18. $\sin^4\frac{\pi}{8} + \sin^4 \frac{3\pi}{8} + \sin^4\frac{5\pi}{8} + \sin^4\frac{7\pi}{8} = \frac{3}{2}$

19. $\left(1 + \cos \frac{\pi}{8}\right)\left(1 + \cos\frac{3\pi}{8}\right)\left(1 + \cos\frac{5\pi}{8}\right)\left(1 + \cos \frac{7\pi}{8}\right) = \frac{1}{8}$

20. Find the value of $\sin \frac{23\pi}{24}$

21. If $A = 112^\circ30',$ find the value of $\sin A$ and $\cos A$

Prove that

1. $\sin^224^\circ - \sin^26^\circ = \frac{1}{8}(\sqrt{5} - 1)$

2. $\tan6^\circ.\tan42^\circ.\tan66^\circ.\tan78^\circ = 1$

3. $\sin47^\circ + \sin61^\circ - \sin 11^\circ - \sin25^\circ = \cos 7^\circ$

4. $\sin 12^\circ\sin48^\circ\sin54^\circ = \frac{1}{8}$

5. $\cot 142\frac{1}{2}^\circ = \sqrt{2} + \sqrt{3} - 2 - \sqrt{6}$

6. $\sin^248^\circ - \cos^212^\circ = -\frac{\sqrt{5} + 1}{8}$

7. $4(\sin 24^\circ + \cos6^\circ) = \sqrt{3} + \sqrt{15}$

8. $\cot6^\circ\cot42^\circ\cot66^\circ\cot78^\circ = 1$

9. $\tan12^\circ\tan24^\circ\tan48^\circ\tan84^\circ = 1$

10. $\sin6^\circ\sin42^\circ\sin66^\circ\sin78^\circ = \frac{1}{16}$

11. $\sin\frac{\pi}{5}\sin\frac{2\pi}{5}\sin\frac{3\pi}{5}\sin\frac{4\pi}{5} = \frac{5}{16}$

12. $\cos36^\circ\cos72^\circ\cos108^\circ\cos144^\circ = \frac{1}{16}$

13. $\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{3\pi}{15}\cos\frac{4\pi}{15}\cos\frac{5\pi}{15}\cos\frac{6\pi}{15}\cos\frac{7\pi}{15} = \frac{1}{2^7}$

14. $\cos\frac{\pi}{65}\cos\frac{2\pi}{65}\cos\frac{4\pi}{65}\cos\frac{8\pi}{65}\cos\frac{16\pi}{65}\cos\frac{32\pi}{65} = \frac{1}{64}$

15. If $\tan \frac{A}{2} = \sqrt{\frac{a - b}{a + b}}\tan \frac{B}{2},$ prove that, $\cos A = \frac{a\cos B + b}{a + b\cos B}$

16. If $\tan \frac{A}{2} \ = \sqrt{\frac{1 - e}{1 + e}}\tan\frac{B}{2},$ prove that $\cos B = \frac{\cos A - e}{1 - e\cos A}$

17. If $\sin A + \sin B = a$ and $\cos A + \cos B = b,$ prove that $\sin(A + B) = \frac{2ab}{a^2 + b^2}$

18. If $\sin A + \sin B = a$ and $\cos A + \cos B = b,$ prove that $\cos(A - B) = \frac{1}{2}(a^2 + b^2 - 2)$

19. If $A$ and $B$ be two different roots of equation $a\cos\theta + b\sin\theta = c,$ prove that

1. $\tan(A + B) = \frac{2ab}{a^2 - b^2}$

2. $\cos(A + B) = \frac{a^2 - b^2}{a^2 + b^2}$

20. If $\cos A + \cos B = \frac{1}{3}$ and $\sin A + \sin B = \frac{1}{4},$ prove that $\cos \frac{A - B}{2} = \pm\frac{5}{24}$

21. If $2\tan \frac{A}{2} = \tan \frac{B}{2},$ prove that $\cos A = \frac{3 + 5\cos B}{5 + 3\cos B}$

22. If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13},$ prove that one value of $\cos \frac{A - B}{2} = \frac{8}{\sqrt{65}}$

23. If $\sec(A + B) + \sec(A - B) = 2\sec A,$ prove that $\cos B = \pm \sqrt{2}\cos \frac{B}{2}$

24. If $\cos \theta = \frac{\cos\alpha\cos\beta}{1 - \sin\alpha\sin\beta},$ prove that one of the values of $\tan \frac{\theta}{2}$