1Which of these expressions is equal to cos 2A: \( \cos^2 A + \sin^2 A \), \( \cos^2 A - \sin^2 A \), \( 1 - 2\cos^2 A \) or \( 2\sin^2 A - 1 \)?Show Answer

2Which of these expressions is equal to sin 3θ: \( 4\sin\theta - 3\sin^3\theta \), \( 4\sin^3\theta - 3\sin\theta \), \( 3\sin\theta - 4\sin^3\theta \) or \( 3\sin\theta - 4\sin^4\theta \)?Show Answer

3If A = 30°, what is the value of sin 2A: 0, 1, \( \frac{1}{2} \) or \( \frac{\sqrt{3}}{2} \)?Show Answer

4If cos 3A = -1, what must A be: 0°, 30°, 15° or 60°?Show Answer

5Which of these is equal to tan 2A: \( \frac{2\tan A}{1 - \tan^2 A} \), \( \frac{2\tan A}{1 + \tan^2 A} \), \( \frac{1 + \tan^2 A}{1 - \tan^2 A} \) or \( \frac{1 - \tan^2 A}{1 + \tan^2 A} \)?Show Answer

6If \( \sin A = \frac{4}{5} \), find sin 2A, cos 2A and tan 2A.Show Answer

7If \( \cos A = \frac{12}{13} \), find sin 2A, cos 2A and tan 2A.Show Answer

8If \( \cos\theta = \frac{4}{5} \), find sin 3θ, cos 3θ and tan 3θ.Show Answer

9If \( \tan y = \frac{1}{2} \), find tan 3y.Show Answer

10If \( \cos 2A = \frac{7}{25} \), find sin A.Show Answer

11If \( \cos 2A = \frac{13}{36} \), find cos A.Show Answer

12If \( \tan 2A = \frac{120}{119} \), find tan A.Show Answer

13Prove that \( \frac{1 + \cos 2A}{\sin 2A} = \cot A \).Show Answer

14Prove that \( \frac{1 - \cos 2A}{\sin 2A} = \tan A \).Show Answer

15Prove that \( \frac{1 - \cos 2A}{1 + \cos 2A} = \tan^2 A \).Show Answer

16Prove that \( \frac{1 - \tan\theta}{1 + \tan\theta} = \frac{1 - \sin 2\theta}{\cos 2\theta} \).Show Answer

17Prove that \( \frac{\cos 2\alpha}{1 + \sin 2\alpha} = \frac{1 - \tan\alpha}{1 + \tan\alpha} \).Show Answer

18Prove that \( \frac{1 + \sin 2\theta - \cos 2\theta}{1 + \sin 2\theta + \cos 2\theta} = \tan\theta \).Show Answer

19Prove that \( \frac{\sin\theta + \sin 2\theta}{1 + \cos\theta + \cos 2\theta} = \tan\theta \).Show Answer

20Prove that tan A + cot A = 2 cosec 2A.Show Answer

21Prove that cot A - tan A = 2 cot 2A.Show Answer

22Prove that \( \tan(45^\circ + A) = \frac{\cos 2A}{1 - \sin 2A} \).Show Answer

23Prove that \( \frac{1 - \sin 2\theta}{\cos 2\theta} = \frac{\cos\theta - \sin\theta}{\cos\theta + \sin\theta} \).Show Answer

24Prove that \( 2\sin^2(45^\circ - A) = 1 - \sin 2A \).Show Answer

25Prove that \( \tan(45^\circ + \theta) = \sec 2\theta + \tan 2\theta \).Show Answer

26Prove that \( 2\cos^2(45^\circ - A) = 1 + \sin 2A \).Show Answer

27Prove that \( \cos^2(45^\circ - \theta) - \sin^2(45^\circ - \theta) = \sin 2\theta \).Show Answer

28Prove that \( \frac{\sin 5A}{\sin A} - \frac{\cos 5A}{\cos A} = 4\cos 2A \).Show Answer

29Prove that \( \frac{1 - \tan^2(45^\circ - A)}{1 + \tan^2(45^\circ - A)} = \sin 2A \).Show Answer

30If \( \cos\theta = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \cos 2\theta = \frac{1}{2}\left(a^2 + \frac{1}{a^2}\right) \).Show Answer

31If \( \sin A = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \cos 2A = -\frac{1}{2}\left(a^2 + \frac{1}{a^2}\right) \).Show Answer

32If \( \sin\beta = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \sin 3\beta = -\frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) \).Show Answer

33If \( \cos\beta = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \cos 3\beta = \frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) \).Show Answer

34Prove that \( (\sin 2A + \sin 2B)^2 + (\cos 2A - \cos 2B)^2 = 4\sin^2(A + B) \).Show Answer

35Prove that \( (\sin 2A - \sin 2B)^2 + (\cos 2A + \cos 2B)^2 = 4\cos^2(A + B) \).Show Answer

36Prove that \( (1 + \sin 2A + \cos 2A)^2 = 4\cos^2 A (1 + \sin 2A) \).Show Answer

37Prove that \( \sin^2 A - \cos^2 A \cos 2B = \sin^2 B - \cos^2 B \cos 2A \).Show Answer

38Prove that \( \cos^2\alpha + \sin^2\alpha \cos 2\beta = \cos^2\beta + \sin^2\beta \cos 2\alpha \).Show Answer

39Prove that \( (2\cos\theta + 1)(2\cos\theta - 1) = 2\cos 2\theta + 1 \).Show Answer

40Prove that \( (2\cos\theta + 1)(2\cos\theta - 1)(2\cos 2\theta - 1) = 2\cos 4\theta + 1 \).Show Answer

41Prove that \( \frac{\cos^3 A + \sin^3 A}{\cos A + \sin A} = 1 - \frac{1}{2}\sin 2A \).Show Answer

42Prove that \( \cos^6\theta - \sin^6\theta = \cos 2\theta\left(1 - \frac{1}{4}\sin^2 2\theta\right) \).Show Answer

43Prove that \( 4(\sin^6\theta + \cos^6\theta) = 4 - 3\sin^2 2\theta \).Show Answer

44Prove that \( \cos^6\theta - \sin^6\theta = \frac{1}{4}(\cos^3 2\theta + 3\cos 2\theta) \).Show Answer

45Prove that \( \cos^6\theta + \sin^6\theta = \frac{1}{8}(5 + 3\cos 4\theta) \).Show Answer

46Prove that \( \sin^8\theta + \cos^8\theta = \frac{1}{8}(8 - 8\sin^2 2\theta + \sin^4 2\theta) \).Show Answer

47Prove that \( \operatorname{cosec} 10^\circ - \sqrt{3}\sec 10^\circ = 4 \).Show Answer

48Prove that \( \sqrt{3}\operatorname{cosec} 20^\circ - \sec 20^\circ = 4 \).Show Answer

49Prove that \( \sec 40^\circ + \sqrt{3}\operatorname{cosec} 40^\circ = 4 \).Show Answer

50Prove that cosec 2A + cot 4A = cot A - cosec 4A.Show Answer

51Prove that cot 8A + cosec 4A = cot 2A - cosec 8A.Show Answer

52Prove that \( \frac{\sec 4A - 1}{\sec 2A - 1} = \tan 4A \cot A \).Show Answer

53Prove that \( \frac{\sec 8A - 1}{\sec 4A - 1} = \tan 8A \cot 2A \).Show Answer

54Prove that \( \frac{1}{\tan 3\theta + \tan\theta} - \frac{1}{\cot 3\theta + \cot\theta} = \cot 4\theta \).Show Answer

55Prove that \( \frac{\cot\theta}{\cot\theta - \cot 3\theta} - \frac{\tan\theta}{\tan 3\theta - \tan\theta} = 1 \).Show Answer

56Prove that \( 4(\cos^3 10^\circ + \sin^3 20^\circ) = 3(\cos 10^\circ + \sin 20^\circ) \).Show Answer

57Prove that \( 4(\cos^3 20^\circ + \sin^3 50^\circ) = 3(\cos 20^\circ + \sin 50^\circ) \).Show Answer

58Prove that \( \cos^3 A \cos 3A + \sin^3 A \sin 3A = \cos^3 2A \).Show Answer

59Prove that \( \tan\theta + \tan(60^\circ + \theta) - \tan(60^\circ - \theta) = 3\tan 3\theta \).Show Answer

60Prove that \( \cot\theta + \cot(60^\circ + \theta) - \cot(60^\circ - \theta) = 3\cot 3\theta \).Show Answer

61Prove that \( \tan\theta + 2\tan 2\theta + 4\tan 4\theta + 8\cot 8\theta = \cot\theta \).Show Answer

62Express cos 4θ in terms of sin θ.Show Answer

63Express cos 5θ in terms of cos θ.Show Answer

64Prove that \( \frac{\cos A}{\cos A - \sin A} - \frac{\cos A}{\cos A + \sin A} = \tan 2A \).Show Answer

65Prove that \( \tan 2\theta + \sec 2\theta = \frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta} \).Show Answer

66Prove that \( \frac{\cot 2\theta + \tan 2\theta}{\cot 2\theta - \tan 2\theta} = \sec 4\theta \).Show Answer

67Prove that \( \frac{\cos 2A}{1 + \sin 2A} = \tan(45^\circ - A) = \frac{1 - \tan A}{1 + \tan A} \).Show Answer

68If 2 tan α = 3 tan β, prove that \( \tan(\alpha + \beta) = \frac{5\sin 2\beta}{5\cos 2\beta - 1} \).Show Answer

69If 2 tan α = 3 tan β, prove that \( \tan(\alpha - \beta) = \frac{\sin 2\beta}{5 - \cos 2\beta} \).Show Answer

70If \( \tan A = \frac{b}{a} \), prove that a cos 2A + b sin 2A = a.Show Answer

71Prove that \( \frac{\sin A - \sqrt{1 + \sin 2A}}{\cos A - \sqrt{1 + \sin 2A}} = \cot A \).Show Answer

72Prove that \( \sin^4 A = \frac{3}{8} - \frac{1}{2}\cos 2A + \frac{1}{8}\cos 4A \).Show Answer

73Prove that \( \sin^4\frac{\pi}{8} + \sin^4\frac{3\pi}{8} + \sin^4\frac{5\pi}{8} + \sin^4\frac{7\pi}{8} = \frac{3}{2} \).Show Answer

74Show with A = 30° that sin 2A is not equal to 2 sin A.Show Answer

75If \( \tan A = \frac{1}{3} \), find sin 2A and cos 2A without finding sin A or cos A.Show Answer

76Derive the formula for cos 3A in terms of cos A.Show Answer

77Use \( \sin 3A = 3\sin A - 4\sin^3 A \) with A = 30° to check the formula.Show Answer

78If \( \tan A = 1 \), find tan 3A and say what angle this suggests.Show Answer