1Which expression is equal to \( \sin(A+B) \): \( \sin A\cos B - \cos A\sin B \), \( \sin A\cos B + \cos A\sin B \), \( \cos A\cos B - \sin A\sin B \) or \( \sin A\cos A + \cos B\sin B \)?Show Answer

2Which expression is equal to \( \cos(A-B) \): \( \sin A\cos B - \cos A\sin B \), \( \sin A\cos B + \cos A\sin B \), \( \cos A\cos B + \sin A\sin B \) or \( \sin A\cos A + \cos B\sin B \)?Show Answer

3If \( A = 30^\circ \) and \( B = 60^\circ \), what is the value of \( \sin(A+B) \): 0, 1, \( \frac12 \) or \( \frac{\sqrt3}{2} \)?Show Answer

4If \( \sin(A+B) = \frac12 \) and \( B = 30^\circ \), what is A: 0°, 30°, 45° or 60°?Show Answer

5Which relation is correct: \( \tan(A+B) = \tan A + \tan B \), \( \tan(A+B) = \frac{\tan A - \tan B}{1 + \tan A\tan B} \), \( \tan(A+B) = \frac{\tan A + \tan B}{1 + \tan A\tan B} \) or \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B} \)?Show Answer

6Without a calculator or table, prove that \( \cos 15^\circ = \frac{\sqrt3 + 1}{2\sqrt2} \).Show Answer

7Without a calculator or table, prove that \( \sin 75^\circ = \frac{\sqrt3 + 1}{2\sqrt2} \).Show Answer

8Without a calculator or table, prove that \( \tan 75^\circ = 2 + \sqrt3 \).Show Answer

9Without a calculator or table, prove that \( \cos 105^\circ = \frac{1 - \sqrt3}{2\sqrt2} \).Show Answer

10Without a calculator or table, prove that \( \tan 105^\circ = -(2 + \sqrt3) \).Show Answer

11Without a calculator or table, prove that \( \cot 15^\circ = 2 + \sqrt3 \).Show Answer

12If \( \sin A = \frac35 \) and \( \cos B = \frac{12}{13} \), find \( \sin(A+B) \).Show Answer

13If \( \sin A = \frac35 \) and \( \cos B = \frac{12}{13} \), find \( \cos(A+B) \).Show Answer

14If \( \sin A = \frac35 \) and \( \cos B = \frac{12}{13} \), find \( \sin(A-B) \).Show Answer

15If \( \sin A = \frac35 \) and \( \cos B = \frac{12}{13} \), find \( \cos(A-B) \).Show Answer

16If \( \sin A = \frac35 \) and \( \cos B = \frac{12}{13} \), find \( \tan(A+B) \).Show Answer

17If \( \sin A = \frac35 \) and \( \cos B = \frac{12}{13} \), find \( \cot(A-B) \).Show Answer

18Prove that \( \sin(A+B) + \sin(A-B) = 2\sin A\cos B \).Show Answer

19Prove that \( \sin(A+B) - \sin(A-B) = 2\cos A\sin B \).Show Answer

20Prove that \( \cos(A+B) + \cos(A-B) = 2\cos A\cos B \).Show Answer

21Prove that \( \cos(A+B) - \cos(A-B) = -2\sin A\sin B \).Show Answer

22Without a calculator or table, prove that \( \sin 75^\circ - \sin 15^\circ = \frac{1}{\sqrt2} \).Show Answer

23Without a calculator or table, prove that \( \cos 15^\circ - \sin 15^\circ = \frac{1}{\sqrt2} \).Show Answer

24Without a calculator or table, prove that \( \sin 105^\circ - \cos 75^\circ = \frac{1}{\sqrt2} \).Show Answer

25Without a calculator or table, prove that \( \cos 105^\circ + \cos 15^\circ = \frac{1}{\sqrt2} \).Show Answer

26Without a calculator or table, prove that \( \tan 15^\circ + \cot 15^\circ = 4 \).Show Answer

27Without a calculator or table, prove that \( \cos 18^\circ - \sin 18^\circ = \sqrt2\sin 27^\circ \).Show Answer

28Without a calculator or table, prove that \( \sin 35^\circ + \cos 35^\circ = \sqrt2\cos 10^\circ \).Show Answer

29Without a calculator or table, prove that \( \sqrt3\cos 20^\circ + \sin 20^\circ = 2\sin 80^\circ \).Show Answer

30Prove that \( \sin(25^\circ + \alpha)\cos(65^\circ - \alpha) + \cos(25^\circ + \alpha)\sin(65^\circ - \alpha) = 1 \).Show Answer

31Prove that \( \cos(25^\circ + \alpha)\cos(65^\circ + \alpha) - \sin(25^\circ + \alpha)\sin(65^\circ + \alpha) = -\sin 2\alpha \).Show Answer

32Prove that \( \cos(60^\circ + \beta) - \cos\beta + \cos(60^\circ - \beta) = 0 \).Show Answer

33Prove that \( \frac{\tan(45^\circ + \phi) - \tan\phi}{1 + \tan(45^\circ + \phi)\tan\phi} = 1 \).Show Answer

34If \( A + B = \left(\frac{\pi}{4}\right)^c \), prove that \( \tan A + \tan B + \tan A\tan B = 1 \).Show Answer

35If \( A + B = \left(\frac{\pi}{4}\right)^c \), prove that \( (1 + \tan A)(1 + \tan B) = 2 \).Show Answer

36If \( \tan X = \frac23 \) and \( \tan Y = \frac15 \), find the angle \( X + Y \).Show Answer

37If \( \tan A = m \) and \( \tan B = \frac{1}{m} \), where m is positive, find the angle \( A + B \).Show Answer

38Prove that \( \frac{\cos 35^\circ - \sin 35^\circ}{\cos 35^\circ + \sin 35^\circ} = \tan 10^\circ \).Show Answer

39Prove that \( \frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ} = \tan 62^\circ \).Show Answer

40Simplify \( \frac{\cos 8^\circ + \sin 8^\circ}{\cos 8^\circ - \sin 8^\circ} \) to a single ratio.Show Answer

41Prove that \( \frac{\cos 10^\circ - \sin 10^\circ}{\cos 10^\circ + \sin 10^\circ} = \tan 35^\circ \).Show Answer

42Prove that \( \frac{\tan 5A - \tan 4A}{1 + \tan 5A\tan 4A} = \tan A \).Show Answer

43Prove that \( \frac{\tan 7A - \tan 4A}{1 + \tan 7A\tan 4A} = \tan 3A \).Show Answer

44Prove that \( \frac{\tan(A+B) - \tan B}{1 + \tan(A+B)\tan B} = \tan A \).Show Answer

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

46Prove that \( \frac{\sin(A+B)}{\sin A\sin B} = \cot B + \cot A \).Show Answer

47Prove that \( \frac{\cos(A+B)}{\sin A\sin B} = \cot A\cot B - 1 \).Show Answer

48Prove that \( \tan 25^\circ + \tan 20^\circ + \tan 25^\circ\tan 20^\circ = 1 \).Show Answer

49Prove that \( 1 - \tan 27^\circ - \tan 18^\circ = \tan 27^\circ\tan 18^\circ \).Show Answer

50Prove that \( \tan 65^\circ - \tan 20^\circ = 1 + \tan 65^\circ\tan 20^\circ \).Show Answer

51Prove that \( 1 + \cot 18^\circ + \cot 27^\circ = \cot 18^\circ\cot 27^\circ \).Show Answer

52Prove that \( \tan 50^\circ + \tan 60^\circ + \tan 70^\circ = \tan 50^\circ\tan 60^\circ\tan 70^\circ \).Show Answer

53Prove that \( \tan 10A - \tan 6A - \tan 4A = \tan 10A\tan 6A\tan 4A \).Show Answer

54Prove that \( \tan 20^\circ\tan 30^\circ + \tan 30^\circ\tan 40^\circ + \tan 40^\circ\tan 20^\circ = 1 \).Show Answer

55Prove that \( \cot 5A\cot 4A - \cot 9A\cot 5A - \cot 9A\cot 4A = 1 \).Show Answer

56Prove that \( \tan 50^\circ - \tan 40^\circ = 2\tan 10^\circ \).Show Answer

57Prove that \( \tan 70^\circ = 2\tan 50^\circ + \tan 20^\circ \).Show Answer

58Prove that \( \frac{\sin(\beta + \gamma)}{\sin(\beta - \gamma)} = \frac{\tan\beta + \tan\gamma}{\tan\beta - \tan\gamma} \).Show Answer

59Prove that \( \frac{\sin(A+B)\sin(A-B)}{\cos^2 A\cos^2 B} = \tan^2 A - \tan^2 B \).Show Answer

60Prove that \( \frac{\sin(A+B) + \sin(A-B)}{\cos(A+B) + \cos(A-B)} = \tan A \).Show Answer

61Prove that \( \frac{\sin(A-B)}{\cos A\cos B} + \frac{\sin(B-C)}{\cos B\cos C} + \frac{\sin(C-A)}{\cos C\cos A} = 0 \).Show Answer

62Prove that \( \sin(A+B+C) = \cos A\cos B\cos C(\tan A + \tan B + \tan C - \tan A\tan B\tan C) \).Show Answer

63Prove that \( \cos(A+B+C) = \cos A\cos B\cos C(1 - \tan A\tan B - \tan B\tan C - \tan C\tan A) \).Show Answer

64Prove that \( \tan(A+B+C) = \frac{\tan A + \tan B + \tan C - \tan A\tan B\tan C}{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A} \).Show Answer

65Why is \( \sin(A+B) \) not equal to \( \sin A + \sin B \)? Show it with one example.Show Answer

66In the unit circle proof, why are the chords PR and QS equal?Show Answer

67Find the exact value of \( \cos 75^\circ \).Show Answer

68If \( \tan A = \frac12 \) and \( \tan B = \frac13 \), show that \( A + B = 45^\circ \).Show Answer

69Prove that \( \sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B \).Show Answer