# 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 $$\tan 72^\circ$$¶

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

$$\tan18^\circ\cot18^\circ = 1\Rightarrow \tan72^\circ = \frac{1}{\tan18^\circ} = \frac{\sqrt{10 + 2\sqrt{5}}}{\sqrt{5} - 1}$$

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

$$\cos 36^\circ = 1 - 2\sin^218^\circ = 1 - 2\left(\frac{\sqrt{5} - 1}{4}\right)^2$$

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

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

### 12.2.9. 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 $$\tan \frac{\theta}{2}$$ is $$\frac{\tan \frac{\alpha}{2} - \tan\frac{\beta}{2}}{1 - \tan\frac{\alpha}{2}\tan\frac{\beta}{2}}$$

25. If $$\tan\alpha = \frac{\sin\theta\sin\phi}{\cos\theta + \cos\phi},$$ prove that one of the values of $$\tan\frac{\alpha}{2}$$ is $$\tan\frac{\theta}{2}\tan\frac{\phi}{2}$$

26. If $$\cos\theta = \frac{\cos\alpha + \cos\beta}{1 + \cos\alpha\cos\beta},$$ prove that one of the values of $$\tan\frac{\theta}{2}$$ is $$\tan\frac{\alpha}{2}\tan\frac{\beta}{2}$$