1. If \(\sin 2\theta\) and \(\cos 2\theta\) are solutions of \(x^{2} + bx - c = 0\), then

1. 1. \(b^{2} + 2c + 1 = 0\)
2. 2. \(b^{2} + 2c - 1 = 0\)
3. 3. \(b^{2} - 2c + 1 = 0\)
4. 4. \(b^{2} - 2c - 1 = 0\)

2. If \(\cot\theta+\tan\theta=3\), and \(1-\cos^{2}\theta-\alpha\cos\theta=0\), then

1. 1. \(6\alpha^{2}(9 - \alpha^{2}) = 1\)
2. 2. \(6\alpha^{2}(\alpha^{2} - 9) = 1\)
3. 3. \(9\alpha^{2}(6 - \alpha^{2}) = 1\)
4. 4. \(9\alpha^{2}(\alpha^{2} - 6) = 1\)

3. If \(\sin\theta+\csc\theta=2\), then \(\sin^{2020}\theta+\csc^{2020}\theta=\)

1. 1. 2
2. 2. \(2020\cdot 2^{2019}\)
3. 3. \(2^{2019}\)
4. 4. 2

4. If \(\sec\theta=m,\tan\theta=n\), then \(\frac{1}{m}\left[\frac{1}{m + n + \frac{1}{m + n}}\right] =\)

1. 1. 1
2. 2. 2
3. 3. -1
4. 4. 3

5. In triangle ABC, if \(\tan A = 2k, \tan B = 3k, \tan C = 4k\), then the value of \(\sec^{2}A+\sec^{2}B+\sec^{2}C=\)

1. 1. \(\frac{101}{8}\)
2. 2. \(\frac{111}{8}\)
3. 3. \(\frac{121}{8}\)
4. 4. \(\frac{91}{8}\)

6. If \(4\cos x+3\sin x=5\), then find the value of \(\tan x=\)

1. 1. \(\frac{3}{4}\)
2. 2. \(\frac{4}{3}\)
3. 3. \(\frac{-3}{4}\)
4. 4. \(\frac{-4}{3}\)

7. The value of \((\sin 210^{\circ})(\sin 585^{\circ})\) is

1. 1. \(\frac{1}{2\sqrt{2}}\)
2. 2. \(\frac{-1}{2\sqrt{2}}\)
3. 3. \(\frac{1}{\sqrt{3}}\)
4. 4. \(\frac{-1}{\sqrt{3}}\)

8. Geometric mean of \(\tan 1^{\circ}\tan 2^{\circ}\ldots\tan 89^{\circ}\) is

1. 1. \(\frac{1}{89}\)
2. 2. 1
3. 3. \(\frac{1}{3}\)
4. 4. \(\sqrt{3}\)

9. \(\sin\left(\frac{5\pi}{3}\right)+\sec\left(\frac{13\pi}{3}\right)=\)

1. 1. \(2 - \frac{\sqrt{3}}{2}\)
2. 2. \(2 + \frac{\sqrt{3}}{2}\)
3. 3. \(\sqrt{3} +\frac{1}{\sqrt{2}}\)
4. 4. \(\sqrt{3} -\frac{1}{\sqrt{2}}\)

10. If \(x\neq 0\), then \(\frac{\sin(\pi + x)\cos(\frac{\pi}{2} + x)\tan(\frac{3\pi}{2} - x)\cot(2\pi - x)}{\sin(2\pi - x)\cos(2\pi + x)\csc(-x)\sin(\frac{3\pi}{2} + x)} =\)

1. 1. 0
2. 2. -1
3. 3. 1
4. 4. 2

11. \(\tan\left(-\frac{23\pi}{3}\right) - \cot\left(\theta -\frac{13\pi}{3}\right) =\)

1. 1. \(\sqrt{3} +\cot\theta\)
2. 2. \(\sqrt{3} -\tan(\frac{\pi}{6} +\theta)\)
3. 3. \(\sqrt{3} +\tan\theta\)
4. 4. \(\sqrt{3} +\cot(\frac{\pi}{3} -\theta)\)

12. If \(\frac{x}{\cos\alpha} = \frac{y}{\cos\left(\frac{2\pi}{3} - \alpha\right)} = \frac{z}{\cos\left(\frac{2\pi}{3} + \alpha\right)}\), then \((x + y + z)\) equals

1. 1. 1/2
2. 2. 0
3. 3. 1
4. 4. 2

13. If \(\sec\theta+\tan\theta=\frac{2}{3}\), then in which quadrant does \(\theta\) lie?

1. 1. I
2. 2. II
3. 3. III
4. 4. IV

14. If \(\csc\theta+\cot\theta=\frac{1}{3}\), then \(\theta\) lies in the

1. 1. \(1^{\text{st}}\) quadrant
2. 2. \(2^{\text{nd}}\) quadrant
3. 3. \(3^{\text{rd}}\) quadrant
4. 4. \(4^{\text{th}}\) quadrant

15. \(\frac{\tan 52^{\circ} - \tan 38^{\circ}}{\tan 14^{\circ}} =\)

1. 1. 1
2. 2. 2
3. 3. \(2\sqrt{3}\)
4. 4. \(\frac{2}{\sqrt{3}}\)

16. \(\cos^{2}\left(\frac{7\pi}{8}\right) + \cos^{2}\left(\frac{5\pi}{8}\right) + \cos^{2}\left(\frac{3\pi}{8}\right) + \cos^{2}\left(\frac{\pi}{8}\right) =\)

1. 1. \(\frac{3}{2}\)
2. 2. \(\frac{2}{3}\)
3. 3. 2
4. 4. 1

17. If \(\theta\) is the angle of a pentagon, then \(\left|(\sin\theta)\hat{i} +(\cos\theta)\hat{j} +(\tan\theta)\hat{k}\right| =\)

1. 1. \(\sec 18^{\circ}\)
2. 2. \(\csc 18^{\circ}\)
3. 3. \(-\sec 18^{\circ}\)
4. 4. \(\csc 108^{\circ}\)

18. If \(A = \sin\theta|\sin\theta|\), \(B = \cos\theta|\cos\theta|\) and \(\frac{99\pi}{2}\leq\theta\leq\frac{100\pi}{2}\), then

1. 1. \(A + B = 1\)
2. 2. \(A + B = -1\)
3. 3. \(B - A = 1\)
4. 4. \(B - A = -1\)

19. \(\sin^{4}\frac{\pi}{8} +\cos^{4}\frac{3\pi}{8} -\sin^{4}\frac{3\pi}{8} +\sin^{4}\frac{5\pi}{8} +\cos^{4}\frac{7\pi}{8} -\sin^{4}\frac{7\pi}{8} =\)

1. 1. \(\frac{1}{4}\)
2. 2. \(\frac{1}{2}\)
3. 3. 0
4. 4. \(\frac{3}{4}\)

20. If \(\alpha = \frac{\sin^{3}x}{\cos^{2}x}\), \(\beta = \frac{\cos^{3}x}{\sin^{2}x}\) and \(\sin x + \cos x = k\), then \(\alpha\sin x + \beta\cos x + 3 =\)

1. 1. \(\frac{2}{(k^{2} - 1)^{3}}\)
2. 2. \(\frac{4}{(k^{2} - 1)^{2}}\)
3. 3. \(\frac{k^{2} - 1}{2}\)
4. 4. \(\frac{(k^{2} - 1)^{2}}{4}\)

21. What is the value of \(\cos\left(22\frac{1}{2}\right)^{\circ} =\)

1. 1. \(\sqrt{\frac{\sqrt{2} - 1}{2\sqrt{2}}}\)
2. 2. \(\sqrt{\frac{\sqrt{2} + 1}{2\sqrt{2}}}\)
3. 3. \(\sqrt{2} - 1\)
4. 4. \(\sqrt{2} + 1\)

22. If \(\cos\theta = -\frac{\sqrt{3}}{2}\) and \(\sin\alpha = -\frac{3}{5}\) where \(\theta\) does not lie in the third quadrant, then the value of \(\frac{2\tan\alpha + \sqrt{3}\tan\theta}{\cot^{2}\theta + \cos\alpha}\) is equal to

1. 1. \(\frac{7}{22}\)
2. 2. \(\frac{5}{22}\)
3. 3. \(\frac{9}{22}\)
4. 4. \(\frac{22}{5}\)

23. Let \(\theta\) be an angle in the standard position such that the point \((-5,12)\) lies on its terminal side, then

1. 1. \(|\sin\theta| = -\sin\theta\)
2. 2. \(|\cos\theta| = \cos\theta\)
3. 3. \(|\tan\theta| = -\tan\theta\)
4. 4. \(|\cos\theta| = -\cos\theta\)

24. Determine the value of 'a' in \(\tan 70^{\circ} - \tan 20^{\circ} = a\cdot\tan 50^{\circ}\)

1. 1. -4
2. 2. 4
3. 3. -2
4. 4. 2

25. \(\sin^{2}5^{\circ} + \sin^{2}10^{\circ} + \sin^{2}15^{\circ} + \ldots +\sin^{2}90^{\circ} =\)

1. 1. \(\frac{8}{2}\)
2. 2. 9
3. 3. \(\frac{9}{2}\)
4. 4. \(\frac{4}{2}\)

26. If \(\sin\theta+\csc\theta = 2\), then the value of \(\sin^{10}\theta+\csc^{10}\theta =\)

1. 1. 2
2. 2. 2
3. 3. \(2^{9}\)
4. 4. \(2^{8}\)

27. Given \(\frac{\sin 1^{\circ}}{\sin x^{\circ}\sin(x + 1)^{\circ}} = \cot x^{\circ} - \cot(x + 1)^{\circ}\), then the value of \(\frac{1}{\sin 45^{\circ}\sin 46^{\circ}} + \frac{1}{\sin 46^{\circ}\sin 47^{\circ}} + \ldots + \frac{1}{\sin 89^{\circ}\sin 90^{\circ}}\) is

1. 1. \(\sin 1^{\circ}\)
2. 2. \(\cot 1^{\circ}\)
3. 3. \(-\cot 1^{\circ}\)
4. 4. \(\csc 1^{\circ}\)

28. \((1 - \tan 348^{\circ})(1 + \cot 417^{\circ}) =\)

1. 1. \(3\sqrt{3}\)
2. 2. 2
3. 3. \(\frac{2}{\sqrt{3}}\)
4. 4. 1

29. If \(0< \theta < \frac{\pi}{2}\) and \(\sin\theta\cos\theta = \frac{12}{25}\), then \(\sin^{4}\theta+\cos^{4}\theta =\)

1. 1. \(\frac{327}{625}\)
2. 2. \(\frac{337}{625}\)
3. 3. \(\frac{347}{625}\)
4. 4. \(\frac{340}{625}\)

30. The value of \((1 - \cos\theta)(1 + \cos\theta)(1 + \cot^{2}\theta)\) when \(\theta = \frac{\pi}{15}\) is

1. 1. 1
2. 2. \(\frac{1}{2}\)
3. 3. \(-\frac{1}{\sqrt{3}}\)
4. 4. 2

31. \(\cot\frac{\pi}{16}\cdot\cot\frac{2\pi}{16}\cdot\cot\frac{3\pi}{16}\cdot\cot\frac{4\pi}{16}\cdot\cot\frac{5\pi}{16}\cdot\cot\frac{6\pi}{16}\cdot\cot\frac{7\pi}{16} =\)

1. 1. 0
2. 2. 1
3. 3. \(\frac{1}{2}\)
4. 4. 2

32. \(2(\sin^{6}\theta+\cos^{6}\theta) - 3(\sin^{4}\theta+\cos^{4}\theta) =\)

1. 1. -1
2. 2. 1
3. 3. 0
4. 4. 12

33. \(\left(\frac{\sin 35^{\circ}}{\cos 55^{\circ}}\right)^{2} + \left(\frac{\cos 55^{\circ}}{\sin 35^{\circ}}\right)^{2} - 2\cos 30^{\circ} =\)

1. 1. \(2 + \sqrt{3}\)
2. 2. \(2 - \sqrt{3}\)
3. 3. \(2\sqrt{3}\)
4. 4. \(3\sqrt{2}\)

34. If \(\frac{2\sin\alpha}{1 + \cos\alpha + \sin\alpha} = x\), then \(\frac{1 - \cos\alpha - \sin\alpha}{\cos\alpha} =\)

1. 1. \(\frac{1}{x}\)
2. 2. \(-x\)
3. 3. \(1 - x\)
4. 4. \(1 + x\)

35. \(\cos^{4}\frac{\pi}{8} + \cos^{4}\frac{2\pi}{8} + \cos^{4}\frac{3\pi}{8} + \cos^{4}\frac{4\pi}{8} + \cos^{4}\frac{5\pi}{8} + \cos^{4}\frac{6\pi}{8} + \cos^{4}\frac{7\pi}{8} + \cos^{4}\frac{8\pi}{8} =\)

1. 1. 3
2. 2. -1
3. 3. 1
4. 4. 4

36. If \((1 + \tan 1^{\circ})(1 + \tan 2^{\circ})\ldots(1 + \tan 45^{\circ}) = 2^{n}\), then \(n =\)

1. 1. 0
2. 2. 32
3. 3. 23
4. 4. 2

37. \(\frac{\cos\theta}{1 - \tan\theta} + \frac{\sin\theta}{1 - \cot\theta} =\)

1. 1. \(\cos\theta - \sin\theta\)
2. 2. \(\sin\theta - \cos\theta\)
3. 3. \(\cos\theta + \sin\theta\)
4. 4. \((1 - \tan\theta)\sin\theta\)

38. If \(A + B + C = \pi\), \(\cos B = \cos A\cos C\), then \(\tan A\tan C =\)

1. 1. 0
2. 2. 1
3. 3. 2
4. 4. 1/2

39. \(1 + \sec^{2}x\sin^{2}x =\)

1. 1. \(\sin 2x\)
2. 2. \(\sin^{2}x\)
3. 3. \(\tan^{2}x\)
4. 4. \(\sec^{2}x\)

40. \(\frac{1}{\sin 1^{\circ}\sin 2^{\circ}} + \frac{1}{\sin 2^{\circ}\sin 3^{\circ}} + \ldots + \frac{1}{\sin 89^{\circ}\sin 90^{\circ}} =\)

1. 1. \(\frac{\cos 1^{\circ}}{\sin 1^{\circ}}\)
2. 2. \(\frac{\cos 1^{\circ}}{\sin^{2}1^{\circ}}\)
3. 3. \(\frac{\sin 1^{\circ}}{\cos 1^{\circ}}\)
4. 4. \(\frac{\sin^{2}1^{\circ}}{\cos 1^{\circ}}\)

41. Which of the following trigonometric values are negative?
I) \(\sin(-292^{\circ})\) II) \(\tan(-103^{\circ})\) III) \(\cos(-207^{\circ})\) IV) \(\cot(-222^{\circ})\)

1. 1. II, III and IV
2. 2. III only
3. 3. I and III
4. 4. II and III

42. If \(\sin\theta+\csc\theta = 4\), then \(\sin^{2}\theta+\csc^{2}\theta =\)

1. 1. 12
2. 2. 18
3. 3. 16
4. 4. 14

43. A true statement among the following identities is

1. 1. \(\cos 5\theta = 16\cos^{5}\theta -20\cos^{3}\theta -5\cos\theta\)
2. 2. \(\cos 5\theta = 20\cos^{3}\theta -16\cos^{5}\theta +5\cos\theta\)
3. 3. \(\cos 5\theta = 16\cos^{5}\theta +20\cos^{3}\theta -5\cos\theta\)
4. 4. \(\cos 5\theta = 16\cos^{5}\theta -20\cos^{3}\theta +5\cos\theta\)

44. If \(\cos\theta - \sin\theta = \sqrt{5}\sin\theta\), then \(\cos\theta + 4\sin\theta =\)

1. 1. \(5\cos\theta\)
2. 2. \(\sqrt{5}\sin\theta\)
3. 3. \(5\sin\theta\)
4. 4. \(\sqrt{5}\cos\theta\)

45. Let a and b be non-negative real numbers. If \(\sin x + a\cos x = b\), then \(|a\sin x - \cos x| =\)

1. 1. \(\sqrt{a^{2} - b^{2} + 1}\)
2. 2. \(\sqrt{b^{2} - a^{2} + 1}\)
3. 3. \(\sqrt{1 + a^{2} + b^{2}}\)
4. 4. \(\sqrt{a^{2} + b^{2} - 1}\)

46. \(\sqrt{\sin^{4}x + 4\cos^{2}x} - \sqrt{\cos^{4}x + 4\sin^{2}x} =\)

1. 1. \(1 - \cos 2x\)
2. 2. \(\tan 2x\)
3. 3. \(\sin 2x\)
4. 4. \(\cos 2x\)

47. \(\frac{1}{1 + \sin\theta} + \frac{1}{1 - \sin\theta} =\)

1. 1. \(2\cos^{2}\theta\)
2. 2. \(-2\cos^{2}\theta\)
3. 3. \(2\tan^{2}\theta\)
4. 4. \(2\sec^{2}\theta\)

48. \(\frac{\cos x}{1 + \sin x} + \tan x =\)

1. 1. 1
2. 2. \(\cos x + \sin x\)
3. 3. \(\sin^{2}x\)
4. 4. \(\sec x\)

49. \(1 + \cot^{2}30^{\circ} - \sec^{2}45^{\circ} =\)

1. 1. \(\frac{1}{4}\)
2. 2. \(\frac{1 - \sqrt{3}}{2}\)
3. 3. 2
4. 4. 0

50. \(\frac{1}{\sin 45^{\circ}\sin 46^{\circ}} + \frac{1}{\sin 47^{\circ}\sin 48^{\circ}} + \ldots + \frac{1}{\sin 133^{\circ}\sin 134^{\circ}} = \frac{1}{\sin(n^{\circ})}\). Then \(n\) is

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

51. \(\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x} =\)

1. 1. \(2\sec x\)
2. 2. \(2\csc x\)
3. 3. \(\tan 2x\)
4. 4. \(\sin 2x\)

52. \(2\cot^{2}\theta - \cot\theta - 3 =\)

1. 1. \((2\cot\theta - 3)(\cot\theta + 1)\)
2. 2. \((2\cot\theta - 1)(\cot\theta + 3)\)
3. 3. \((2\cot\theta + 3)(\cot\theta - 1)\)
4. 4. \((2\cot\theta + 1)(\cot\theta - 3)\)

53. \(\cos\theta(\csc\theta - \sec\theta) - \cot\theta =\)

1. 1. -1
2. 2. 1
3. 3. 0
4. 4. \(\cos^{2}\theta - \tan^{2}\theta\)

54. \(\tan x + \frac{\cos x}{1 + \sin x} =\)

1. 1. \(\tan 2x\)
2. 2. \(\csc x\)
3. 3. \(\sec x\)
4. 4. \(\cos 2x\)

55. If \(\tan 15^{\circ}\) and \(\tan 30^{\circ}\) are the roots of the equation \(x^{2} + px + q = 0\), then \(pq =\)

1. 1. \(\frac{6\sqrt{3} + 10}{\sqrt{3}}\)
2. 2. \(\frac{10 - 6\sqrt{3}}{3}\)
3. 3. \(\frac{10 + 6\sqrt{3}}{3}\)
4. 4. \(\frac{10 - 6\sqrt{3}}{\sqrt{3}}\)

56. If \(\frac{1}{\sin 45^{\circ}\sin 46^{\circ}} + \frac{1}{\sin 46^{\circ}\sin 47^{\circ}} + \ldots\) upto 45 terms \(= \frac{1}{\sin x^{\circ}}\), then \(\sin\left(\frac{\pi}{2}x\right) =\)

1. 1. 0
2. 2. \(\sin 1\)
3. 3. 1
4. 4. \(\cos 1\)

57. If \(\sin A = -\frac{7}{25}\), \(\cos B = \frac{8}{17}\), A does not lie in the \(3^{\text{rd}}\) quadrant and B does not lie in the \(1^{\text{st}}\) quadrant, then \(8\tan A - 5\cot B =\)

1. 1. 0
2. 2. \(\frac{1}{3}\)
3. 3. \(\frac{1}{2}\)
4. 4. 1

58. \(\sin 21^{\circ}\cos 9^{\circ} - \cos 84^{\circ}\cos 6^{\circ} =\)

1. 1. 1
2. 2. \(\frac{1}{4}\)
3. 3. \(\frac{1}{2}\)
4. 4. \(\frac{3}{2}\)

59. If \(1 + \sqrt{1 + a} = (1 + \sqrt{1 - a})\cot\alpha\) and \(0< a < 1\), then \(\sin 4\alpha =\)

1. 1. a
2. 2. 2a
3. 3. 3a
4. 4. 4a

60. If \(A = \frac{\pi}{24}\), then \(\frac{\cos A + \cos 3A + \cos 5A + \cos 7A}{\sin A + \sin 3A + \sin 5A + \sin 7A} =\)

1. 1. \(\sqrt{3}\)
2. 2. \(\sqrt{3}\)
3. 3. \(\frac{1}{\sqrt{3}}\)
4. 4. \(\frac{2}{\sqrt{3}}\)

61. If \(\sec(\theta+\alpha)\), \(\sec\theta\) and \(\sec(\theta-\alpha)\) are in arithmetic progression, then \(\sin^{2}\theta =\)

1. 1. \(\cos\alpha\)
2. 2. \(-\cos\alpha\)
3. 3. \(-2\cos\alpha\)
4. 4. \(-\cos\alpha\)

62. If \(\cos\alpha+\cos\beta = a\) and \(\sin\alpha+\sin\beta = b\), then match the items given in List-A with those of their values in List-B.

1. 1. (I)→(a) (II)→(e) (III)→(d) (IV)→(c)
2. 2. (I)→(a) (II)→(c) (III)→(b) (IV)→(e)
3. 3. (I)→(a) (II)→(d) (III)→(c) (IV)→(b)
4. 4. (I)→(a) (II)→(d) (III)→(b) (IV)→(c)

63. If \(\tan A+ \tan B=x\) and \(\cot A+ \cot B=y\), then \(\tan(A+B)=\)

1. 1. \(\frac{xy}{x - y}\)
2. 2. \(\frac{xy}{y - x}\)
3. 3. \(\frac{xy}{x + y}\)
4. 4. \(\frac{x - y}{xy}\)

64. If \(\left[1 - \cos\left(\frac{\pi}{2} + \alpha\right) + \sin\left(\frac{3\pi}{2} + \alpha\right)\right]^{2} + \left[1 - \sin\left(\frac{3\pi}{2} - \alpha\right) - \cos\left(\frac{3\pi}{2} + \alpha\right)\right]^{2} = a + b\sin^{2}\left(\frac{\pi}{4} + \alpha\right)\), then \(a^{2} + b^{2} =\)

1. 1. 20
2. 2. 52
3. 3. 40
4. 4. 32

65. \(\frac{\cot A}{1 - \tan A} + \frac{\tan A}{1 - \cot A} =\)

1. 1. \(\tan A+ \cot A\)
2. 2. \(\sec A+ \csc A\)
3. 3. \(\sin A\cos A+ 1\)
4. 4. \(\sec A\csc A+ 1\)

66. \(\frac{\tan A+ \cot A}{1 - \cot A} =\)

1. 1. \(\sec A\csc A- 1\)
2. 2. \(\tan A+ \cot A\)
3. 3. \(\tan A+ \cot A+ 1\)
4. 4. \(\sec A+ \csc A+ 1\)

67. If \(10\sin^{4}\alpha + 15\cos^{4}\alpha = 6\), then \(16\tan^{6}\alpha + 27\cot^{6}\alpha =\)

1. 1. 43
2. 2. 54
3. 3. 62
4. 4. 59

68. \(\sum_{k=0}^{4}\sin^{2}(2k + 1)\frac{\pi}{20} =\)

1. 1. 5
2. 2. \(\frac{5}{2}\)
3. 3. 3
4. 4. 3

69. \(\frac{1 + \cos\theta - \sin\theta}{1 + \cos\theta + \sin\theta} + \frac{1 + \cos\theta + \sin\theta}{1 + \cos\theta - \sin\theta} =\)

1. 1. \(2\sec\theta\)
2. 2. \(2\csc\theta\)
3. 3. \(2\tan\theta\)
4. 4. \(2\cot\theta\)

70. If \(f_{n}(x) = \frac{1}{2n}[\sin^{2n}x + \cos^{2n}x]\), then \(f_{1}(x) + f_{2}(x) - f_{3}(x) =\)

1. 1. 0
2. 2. \(\frac{5}{12}\)
3. 3. \(\frac{11}{12}\)
4. 4. \(\frac{7}{12}\)

71. Match the items of List-A with those of the entries of List-B.

1. 1. (I)→(B), (II)→(D), (III)→(C), (IV)→(A)
2. 2. (I)→(B), (II)→(E), (III)→(A), (IV)→(C)
3. 3. (I)→(B), (II)→(C), (III)→(A), (IV)→(D)
4. 4. (I)→(C), (II)→(B), (III)→(D), (IV)→(E)

72. \(\sin\alpha+\cos\alpha = m \Rightarrow \sin^{6}\alpha+\cos^{6}\alpha =\)

1. 1. \(\frac{4 + 3(m^{2} - 1)^{2}}{4}\)
2. 2. \(\frac{4 - 3(m^{2} - 1)^{2}}{4}\)
3. 3. \(\frac{3 + 4(m^{2} - 1)^{2}}{4}\)
4. 4. \(\frac{4 - 3(m^{2} + 1)^{2}}{4}\)

73. If \(\frac{2\sin\theta}{1 + \cos\theta + \sin\theta} = y\), then \(\frac{1 - \cos\theta + \sin\theta}{1 + \sin\theta} =\)

1. 1. y
2. 2. \(\frac{1}{y}\)
3. 3. \(1 - y\)
4. 4. \(1 + y\)

74. If \(\cot\theta = -\frac{2}{3}\) and \(\theta\) does not lie in the \(4^{\text{th}}\) quadrant, then \(\frac{(5\sin\theta + \cos\theta)^{2}}{\tan\theta + \cot\theta} =\)

1. 1. \(\frac{13}{13}\)
2. 2. -6
3. 3. \(\frac{1734}{169}\)
4. 4. 13

1. Let \(\alpha,\beta,\gamma\) be such that \(0<\alpha<\beta<\gamma<2\pi\). For any \(x\in\mathbb{R}\), if \(\cos(x+\alpha)+\cos(x+\beta)+\cos(x+\gamma)=0\), then \(\tan(\gamma-\alpha)=\)

1. 1. \(\sqrt{3}\)
2. 2. 0
3. 3. 1
4. 4. \(\sqrt{3}\)

2. If ABC is not a right-angled triangle and \(\sin\left(\frac{\pi}{4}-A\right)\sin\left(\frac{\pi}{4}-B\right)=-\frac{1}{2\sqrt{2}}\cos\left(\frac{\pi}{4}-C\right)\), then \(\tan A\tan B+\tan B\tan C+\tan C\tan A=\)

1. 1. \(\cot A+\cot B+\cot C\)
2. 2. \(\tan A+\tan B+\tan C\)
3. 3. \(\frac{1}{\tan A+\tan B+\tan C}\)
4. 4. \(\frac{1}{\cot A+\cot B+\cot C}\)

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

1. 1. \(\frac{3}{2}\)
2. 2. \(\frac{1}{2}\)
3. 3. \(\frac{-3}{2}\)
4. 4. \(\frac{-1}{2}\)

4. \(\sqrt{3}\sin(\theta)+\cos(\theta)=2\sin\left(\theta+\frac{\pi}{6}\right)\)

1. 1. -2
2. 2. 1
3. 3. 2
4. 4. -1

5. If \(\tan\alpha=2\sin\beta\sin\gamma\csc(\beta+\gamma)\), then

1. 1. \(\cot\beta,\cot\alpha,\cot\gamma\) are in HP
2. 2. \(\tan\gamma,\tan\alpha,\tan\beta\) are in HP
3. 3. \(\cot\alpha,\cot\beta,\cot\gamma\) are in AP
4. 4. \(\tan\alpha,\tan\beta,\tan\gamma\) are in AP

6. Assertion (A): If \(A=15^{\circ},B=17^{\circ}\) and \(C=13^{\circ}\), then \(\cot2A+\cot2B+\cot2C=\cot2A\cot2B\cot2C\).
Reason (R): In a \(\Delta PQR\), \(\tan\frac{P}{2}\tan\frac{Q}{2}+\tan\frac{Q}{2}\tan\frac{R}{2}+\tan\frac{R}{2}\tan\frac{P}{2}=1\).

1. 1. (A) true, (R) true, (R) correct explanation
2. 2. (A) true, (R) true, (R) not correct explanation
3. 3. (A) true, (R) false
4. 4. (A) false, (R) true

7. In a triangle ABC, if \(\cos A\cos B+\sin A\sin B\sin C=1\), then \(a:b:c=\)

1. 1. \(1:1:\sqrt{2}\)
2. 2. \(1:1:1\)
3. 3. \(\sqrt{2}:1:1\)
4. 4. \(1:\sqrt{2}:1\)

8. If A does not belong to the first quadrant, B does not belong to the second quadrant, \(\sin A=\frac{11}{61}\) and \(\cos B=\frac{-7}{25}\), then \(A-B\) and \(A+B\) lie respectively in the quadrants

1. 1. 2, 2
2. 2. 3, 1
3. 3. 4, 1
4. 4. 1, 4

9. In a triangle ABC, if \(3\sin A+4\cos B=6\) and \(4\sin B+3\cos A=1\), then \(\sin(A+B)=\)

1. 1. 1
2. 2. \(\frac{1}{2}\)
3. 3. 0
4. 4. \(\cos C\)

10. If \(f(x)=\frac{\cot x}{1+\cot x}\) and \(\alpha+\beta=\frac{5\pi}{4}\), then \(f(\alpha)f(\beta)=\)

1. 1. \(\frac{3}{2}\)
2. 2. \(\frac{-3}{2}\)
3. 3. \(\frac{-1}{2}\)
4. 4. \(\frac{1}{2}\)

11. If \(x\cos\theta=y\cos\left(\theta+\frac{2\pi}{3}\right)=z\cos\left(\theta+\frac{4\pi}{3}\right)\), then \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\)

1. 1. 1
2. 2. 2
3. 3. 0
4. 4. 3

12. If \(\cos\frac{\pi}{4}\cos\frac{\pi}{8}\cos\frac{\pi}{16}\cos\frac{\pi}{32}=2^{m}\csc\frac{\pi}{n}\), then \(m+n=\)

1. 1. 27
2. 2. 25
3. 3. 28
4. 4. 29

13. \(\sin\frac{2\pi}{5}+\sin\frac{4\pi}{5}+\sin\frac{6\pi}{5}+\sin\frac{8\pi}{5}=\)

1. 1. 0
2. 2. 1
3. 3. \(\frac{\sqrt{2}}{2}\)
4. 4. \(\frac{1}{2}\)

14. \(\sin\frac{\pi}{16}\sin\frac{3\pi}{16}\sin\frac{5\pi}{16}\sin\frac{7\pi}{16}=\)

1. 1. \(\frac{\sqrt{2}}{16}\)
2. 2. \(\frac{1}{8}\)
3. 3. \(\frac{1}{16}\)
4. 4. \(\frac{\sqrt{2}}{32}\)

15. \(\cos^{2}10^{\circ}+\cos^{2}50^{\circ}-\sin40^{\circ}\sin80^{\circ}=\)

1. 1. \(\frac{1}{4}\)
2. 2. \(\frac{1}{2}\)
3. 3. \(\frac{4}{3}\)
4. 4. \(\frac{3}{4}\)

16. If \(\alpha+\beta=\gamma\), then \(\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=\)

1. 1. \(1+2\cos\alpha\cos\beta\cos\gamma\)
2. 2. \(1+2\cos^2\alpha\cos^2\beta\cos^2\gamma\)
3. 3. \(1+2\cos\alpha\cos\beta\cos\gamma\)
4. 4. \(1+4\cos\alpha\cos\beta\cos\gamma\)

17. \(\frac{\cot^{2}15^{\circ}-1}{\cot^{2}15^{\circ}+1}=\)

1. 1. \(\frac{1}{2}\)
2. 2. \(\frac{\sqrt{3}}{2}\)
3. 3. \(\frac{3\sqrt{3}}{4}\)
4. 4. \(\frac{\sqrt{3}}{4}\)

18. Let ACB be a triangle with right angle at C. Let AB=29 units, BC=21 units and \(\angle ABC=\theta\). Then \(\cos^{2}\theta-\sin^{2}\theta=\)

1. 1. 1
2. 2. \(\frac{41}{841}\)
3. 3. \(\frac{40}{441}\)
4. 4. \(\frac{41}{800}\)

19. \(\sin20^{\circ}\sin40^{\circ}\sin60^{\circ}\sin80^{\circ}=\)

1. 1. \(\frac{-3}{16}\)
2. 2. \(\frac{5}{16}\)
3. 3. \(\frac{3}{16}\)
4. 4. \(\frac{-5}{16}\)

20. \(\cos\frac{2\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{6\pi}{7}+\cos\frac{7\pi}{7}=\)

1. 1. \(\frac{1}{2}\)
2. 2. 1
3. 3. \(\frac{-1}{2}\)
4. 4. \(\frac{-3}{2}\)

21. If \(\cot A=\frac{11}{60}\), \(\cos B=\frac{7}{25}\) and neither A nor B in the first quadrant, then \(\left(A+\frac{B}{2}\right)\) lies in the quadrant

1. 1. I
2. 2. II
3. 3. III
4. 4. IV

22. \(\sqrt{3}\csc20^{\circ}-\sec20^{\circ}=\)

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

23. The value of \(\tan\left(\frac{7\pi}{8}\right)\) is

1. 1. \(\sqrt{2}-1\)
2. 2. \(1-\sqrt{2}\)
3. 3. \(1+\sqrt{2}\)
4. 4. \(\frac{1}{1+\sqrt{2}}\)

24. \(\sec^{2}x+5\tan x+5=\)

1. 1. \((\tan x+2)(\tan x+3)\)
2. 2. \((\tan x+1)(\tan x+5)\)
3. 3. \((\tan x-2)(\tan x-3)\)
4. 4. \((\sin x+2)(\sin x+5)\)

25. The value of \(\cos^{4}x\) is

1. 1. \(\frac{3}{8}+\frac{1}{2}\cos2x+\frac{1}{8}\cos4x\)
2. 2. \(\frac{3}{8}-\frac{1}{2}\cos2x+\frac{1}{8}\cos4x\)
3. 3. \(\frac{3}{8}-\frac{1}{8}\cos4x+\frac{1}{2}\cos2x\)
4. 4. \(\frac{1}{8}\cos4x+\frac{1}{2}\cos2x-\frac{3}{8}\)

26. \(\sin22\frac{1}{2}^{\circ}=\)

1. 1. \(\sqrt{\frac{2+\sqrt{2}}{4}}\)
2. 2. \(\frac{2+\sqrt{2}}{4}\)
3. 3. \(\sqrt{\frac{2-\sqrt{2}}{4}}\)
4. 4. \(\frac{2-\sqrt{2}}{4}\)

27. \(\cos^{2}45^{\circ}+\cos^{2}135^{\circ}+\cos^{2}225^{\circ}+\cos^{2}315^{\circ}=\)

1. 1. 1
2. 2. 2
3. 3. 0
4. 4. -1

28. \(\sin(x+y)\sec x\sec y=\)

1. 1. \(\cos x\cos y\)
2. 2. \(\tan x-\tan y\)
3. 3. \(\cos x+\cos y\)
4. 4. \(\tan x+\tan y\)

29. In \(\Delta ABC\), if \(3\sin A+4\cos B=6\) and \(4\sin B+3\cos A=1\), then the angle C is

1. 1. \(\frac{\pi}{2}\)
2. 2. \(\frac{\pi}{3}\)
3. 3. \(\frac{\pi}{4}\)
4. 4. \(\frac{\pi}{6}\)

30. The value of \(\sin\left(\frac{5\pi}{24}\right)\cos\left(\frac{\pi}{24}\right)\) is

1. 1. \(\frac{1+\sqrt{2}}{4}\)
2. 2. \(1+\sqrt{2}\)
3. 3. \(\frac{1-\sqrt{2}}{4}\)
4. 4. \(1-\sqrt{2}\)

31. \(\cos\frac{\pi}{12}=\)

1. 1. \(\frac{\sqrt{2}-\sqrt{3}}{2}\)
2. 2. \(\frac{\sqrt{2}+\sqrt{3}}{2}\)
3. 3. \(\frac{\sqrt{2}-\sqrt{6}}{4}\)
4. 4. \(\frac{\sqrt{2}+\sqrt{6}}{4}\)

32. The value of \(\cos\left(\frac{7\pi}{12}\right)\) is

1. 1. \(\frac{\sqrt{2}+\sqrt{3}}{4}\)
2. 2. \(\frac{\sqrt{2}-\sqrt{3}}{4}\)
3. 3. \(\frac{\sqrt{2}-\sqrt{6}}{4}\)
4. 4. \(\frac{\sqrt{2}+\sqrt{6}}{4}\)

33. Let \(\tan30^{\circ}\) and \(\tan15^{\circ}\) be the roots of the quadratic equation \(x^{2}+ax+b=0\), then \(1+a-b=\)

1. 1. 0
2. 2. 1
3. 3. ab
4. 4. \(a^{2}b^{2}\)

34. If \(1-\cot23^{\circ}=\frac{x}{1-\cot22^{\circ}}\), then \(x=\)

1. 1. 1
2. 2. 2
3. 3. \(\frac{1}{2}\)
4. 4. 3

35. If A and B (A>B) are acute angles, \(\sin(A-B)=\frac{16}{65}\) and \(\sin B=\frac{5}{13}\), then \(\tan A+\cot A=\)

1. 1. \(\frac{25}{12}\)
2. 2. \(\frac{12}{25}\)
3. 3. \(\frac{5}{12}\)
4. 4. \(\frac{12}{5}\)

36. If \(\cos x+\cos y=p\), \(\sin x+\sin y=q\), then \(\cos\left(\frac{x-y}{2}\right)=\)

1. 1. \(\pm\frac{\sqrt{p^{2}+q^{2}}}{2}\)
2. 2. \(\pm\frac{pq}{2}\)
3. 3. \(\pm\left(\frac{p+q}{2}\right)\)
4. 4. \(\pm\frac{\sqrt{p^{2}+q^{2}}}{4}\)

37. If \(\sin(A+B)\sin(A-B)+\cos(A+B)\cos(A-B)=1\) and \(0 [TS EAMCET 20-07-2022_Shift-1]

1. 1. \(\frac{\pi}{6}\)
2. 2. \(\frac{\pi}{4}\)
3. 3. \(\frac{\pi}{3}\)
4. 4. \(\frac{5\pi}{12}\)

38. \(\frac{1}{\cos290^{\circ}}+\frac{1}{\sqrt{3}\sin250^{\circ}}=\)

1. 1. \(\frac{\sqrt{3}}{4}\)
2. 2. \(\frac{4}{\sqrt{3}}\)
3. 3. \(\frac{2}{\sqrt{3}}\)
4. 4. \(\frac{\sqrt{3}}{2}\)

39. In \(\Delta ABC\), if \(\cos A\cos B\cos C=\frac{1}{5}\), then \(\tan A\tan B+\tan B\tan C+\tan C\tan A=\)

1. 1. 4
2. 2. \(\frac{11}{5}\)
3. 3. 6
4. 4. \(\frac{6}{5}\)

40. If \(\cos(\theta-\alpha)\), \(\cos\theta\) and \(\cos(\theta+\alpha)\) are in harmonic progression, then \(2\tan^{2}\theta=\)

1. 1. \(\tan\frac{\alpha}{2}-1\)
2. 2. \(1+\tan\frac{\alpha}{2}\)
3. 3. \(1+\cot\frac{\alpha}{2}\)
4. 4. \(1-\cot\frac{\alpha}{2}\)

41. If \(\cos A+\cos(A+B)+\cos(A-2B)+\ldots\) upto n terms \(=\frac{\cos\left(\frac{2A+(n-1)B}{2}\right)\sin\frac{nB}{2}}{\sin\frac{B}{2}}\), then \(\cos\frac{\pi}{19}+\cos\frac{3\pi}{19}+\ldots+\cos\frac{17\pi}{19}=\)

1. 1. 1
2. 2. \(\frac{1}{2}\)
3. 3. \(\frac{1}{2}\)
4. 4. 0

42. In \(\Delta ABC\), \((\cot A+\cot B)(\cot B+\cot C)(\cot C+\cot A)=\)

1. 1. \(\sec A\sec B\sec C\)
2. 2. \(\tan A\tan B\tan C\)
3. 3. \(\csc A\csc B\csc C\)
4. 4. \(\cot A\cot B\cot C\)

43. If \(\alpha,\beta\) are acute angles such that \(\sin\beta=2\sin\alpha\) and \(3\cos\beta=2\cos\alpha\), then \(\sec(\alpha+\beta)=\)

1. 1. 4
2. 2. \(\sqrt{15}\)
3. 3. \(\sqrt{20}\)
4. 4. 5

44. If \(\tan B=\frac{2\sin A\sin C}{\sin(A+C)}\), then \(\tan A,\tan B,\tan C\) are in

1. 1. AP
2. 2. HP
3. 3. GP
4. 4. AGP

45. If \(\cos\theta,\sin\theta,\cot\theta\) are in GP, then \(\sin^{6}\theta+3\sin^{4}\theta+3\sin^{2}\theta+1=\)

1. 1. 2
2. 2. 7
3. 3. 1
4. 4. 5

46. If \(P=\tan15^{\circ}+\cot15^{\circ}\), \(Q=\tan22^{\circ}+\cot22^{\circ}\) and \(R=\sin54^{\circ}+\sin18^{\circ}\), then their ascending order is

1. 1. P, Q, R
2. 2. P, R, Q
3. 3. R, Q, P
4. 4. R, P, Q

47. \(\frac{1+\tan32^{\circ}}{1-\tan148^{\circ}}=\)

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

1. \(\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}=\)

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

2. If \(\frac{1}{2}\left(\tan\left(\frac{\pi}{24}\right)+\cot\left(\frac{\pi}{24}\right)\right)=\sqrt{a^{2}+a}+\sqrt{a}\), then \(a=\)

1. 1. 3
2. 2. 2
3. 3. 1
4. 4. 4

3. \(\frac{1-\cos(2x)+\sin(x)}{\sin(2x)+\cos(x)}=\)

1. 1. \(\sin(x)\)
2. 2. \(\cos(x)\)
3. 3. \(\tan(x)\)
4. 4. \(\csc(x)\)

4. The value of \(x\) in \(\left(0,\frac{\pi}{2}\right)\) satisfying \((\sin x)(\cos x)=\frac{1}{4}\) is

1. 1. \(\frac{\pi}{6}\)
2. 2. \(\frac{\pi}{3}\)
3. 3. \(\frac{\pi}{8}\)
4. 4. \(\frac{\pi}{12}\)

5. If \(\tan\left(\frac{x}{2}\right)=\frac{m}{n}\), then \(m\sin(x)+n\cos(x)=\)

1. 1. m
2. 2. -m
3. 3. -n
4. 4. n

6. If \(\sin A+\sin B=\frac{1}{2}\) and \(\cos A+\cos B=1\), then \(\sin\left(\frac{A-B}{2}\right)=\)

1. 1. \(\pm\frac{\sqrt{13}}{4}\)
2. 2. \(\pm\frac{\sqrt{11}}{4}\)
3. 3. \(\pm\frac{\sqrt{7}}{4}\)
4. 4. \(\pm\frac{\sqrt{17}}{4}\)

7. If \(\cos(\theta_{1})+\cos(\theta_{2})+\cos(\theta_{3})+\cos(\theta_{4})=-4\), then \(\cot\left(\frac{\theta_{1}}{2}\right)+\cot\left(\frac{\theta_{2}}{2}\right)+\cot\left(\frac{\theta_{3}}{2}\right)+\cot\left(\frac{\theta_{4}}{2}\right)=\)

1. 1. 4
2. 2. 1
3. 3. 2
4. 4. 0

8. \(\tan\left(\frac{3\pi}{16}\right)+\cot\left(\frac{3\pi}{16}\right)=\)

1. 1. \(\sqrt{\sqrt{2}-1}\)
2. 2. \(2\sqrt{\sqrt{2}-1}\)
3. 3. \(2^{3/4}\sqrt{\sqrt{2}-1}\)
4. 4. \(2^{3/4}\sqrt{\sqrt{2}-1}\)

9. If \(\alpha\) is a root of \(25\cos^{2}\theta+5\cos\theta-12=0\) for \(\frac{\pi}{2}<\alpha<\pi\), then \(\sin2\alpha=\)

1. 1. \(\frac{-3}{5}\)
2. 2. \(\frac{-24}{25}\)
3. 3. \(\frac{-4}{25}\)
4. 4. \(\frac{-13}{18}\)

10. \(\csc^{-1}\left[\frac{\tan^{2}\left(\frac{\alpha-\pi}{4}\right)-1}{\tan^{2}\left(\frac{\alpha-\pi}{4}\right)+1}+\cos\frac{\alpha}{2}\cdot\cot5\alpha\right]\sec\frac{11\alpha}{2}=\)

1. 1. \(2\alpha\)
2. 2. \(5\alpha\)
3. 3. \(\frac{\pi}{2}-4\alpha\)
4. 4. \(\frac{5}{2}\alpha\)

11. \(\tan2\alpha\tan(30^{\circ}-\alpha)+\tan2\alpha\tan(60^{\circ}-\alpha)+\tan(60^{\circ}-\alpha)\tan(30^{\circ}-\alpha)\) is equal to

1. 1. \(\tan3\alpha\)
2. 2. \(\tan^{2}2\alpha-\tan^{2}60^{\circ}\)
3. 3. 1
4. 4. 0

12. \(\tan\alpha+2\tan2\alpha+4\tan4\alpha+8\cot8\alpha=\)

1. 1. \(\tan16\alpha\)
2. 2. 0
3. 3. \(\cot\alpha\)
4. 4. \(\tan\alpha\)

13. In a triangle ABC, suppose none of the angles are multiples of \(\frac{\pi}{2}\), then \(\cot A\cot B+\cot B\cot C+\cot A\cot C=\)

1. 1. 2
2. 2. 1
3. 3. -1
4. 4. 0

14. If \(\alpha=\frac{180^{\circ}}{7}\), then \(3\sin\alpha-4\sin^{3}\alpha\) is equal to

1. 1. \(\cot4\alpha\)
2. 2. \(\sin4\alpha\)
3. 3. \(\cos3\alpha\)
4. 4. 0

15. In a triangle \(\Delta ABC\), if \(\tan(A/2),\tan(B/2),\tan(C/2)\) are in Arithmetic progression, then which of the following is always correct?

1. 1. \(\cos A,\cos B,\cos C\) are in AP
2. 2. \(\cos A,\cos B,\cos C\) are in GP
3. 3. \(\cos A,\cos B,\cos C\) are in HP
4. 4. No conclusion can be made

16. If \(90^{\circ} [TS EAMCET 04-08-2021_Shift-2]

1. 1. \(\frac{1}{2}\)
2. 2. \(\frac{3}{5}\)
3. 3. \(\frac{3}{2}\)
4. 4. 2

17. If \(\cos\theta=\frac{-3}{5}\) and \(\pi<\theta<3\pi/2\), then \(\tan\left(\frac{\theta}{2}\right)=\)

1. 1. 2
2. 2. -2
3. 3. 1
4. 4. -1

18. \(\frac{1-\tan^{2}15^{\circ}}{1+\tan^{2}15^{\circ}}=\)

1. 1. 1
2. 2. \(\sqrt{3}\)
3. 3. \(\frac{\sqrt{3}}{2}\)
4. 4. 2

19. \(\frac{1-\cos2\theta+\sin2\theta}{1+\cos2\theta+\sin2\theta}=\)

1. 1. \(\cot\theta\)
2. 2. \(\cos2\theta\)
3. 3. \(\tan\theta\)
4. 4. \(\tan2\theta\)

20. If A is not an integral multiple of \(\frac{\pi}{2}\), then \(\csc2A+\cot2A=\)

1. 1. \(\tan A\)
2. 2. \(\cot A+2\cot2A\)
3. 3. \(\tan A+2\cot2A\)
4. 4. \(\tan2A\)

21. If \(\cos^{4}\theta=a\cos4\theta+b\cos2\theta+c\) for some \(a,b,c\in\mathbb{Q}\), then \((a,b,c)=\)

1. 1. \(\left(\frac{1}{8},\frac{1}{2},\frac{3}{8}\right)\)
2. 2. \(\left(\frac{1}{4},\frac{1}{2},\frac{1}{4}\right)\)
3. 3. \(\left(\frac{1}{8},\frac{1}{4},\frac{3}{8}\right)\)
4. 4. \(\left(\frac{1}{4},\frac{1}{4},\frac{1}{2}\right)\)

22. A true statement among the following identities is

1. 1. \(\sin5\theta=16\cos^{4}\theta\sin\theta-12\cos^{2}\theta\sin\theta+\sin\theta\)
2. 2. \(\sin5\theta=16\cos^{4}\theta-12\cos^{2}\theta+1\)
3. 3. \(\sin5\theta=16\cos^{4}\theta\sin\theta+12\cos^{2}\theta\sin\theta-\sin\theta\)
4. 4. \(\sin5\theta=16\cos^{4}\theta\sin\theta-12\cos^{2}\theta\sin\theta+\sin\theta\)

23. In a triangle ABC, \(\left(\tan\frac{A}{2}\tan\frac{B}{2}\tan\frac{C}{2}\right)^{2}\leq\)

1. 1. \(\frac{1}{27}\)
2. 2. \(\frac{1}{9}\)
3. 3. \(\frac{1}{3}\)
4. 4. 1

24. If \(\sin^{4}\theta\cos^{2}\theta=\sum_{n=0}^{\infty}a_{2n}\cos2n\theta\), then the least \(n\) for which \(a_{2n}=0\) is

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

25. If \(\sin\theta=-\frac{3}{4}\), then \(\sin2\theta=\)

1. 1. \(\frac{3\sqrt{7}}{8}\)
2. 2. \(-\frac{3\sqrt{7}}{8}\)
3. 3. \(\frac{2\sqrt{3}}{7}\)
4. 4. \(\frac{3\sqrt{7}}{8}\)

26. \(\sin^{2}\frac{2\pi}{3}+\cos^{2}\frac{5\pi}{6}-\tan^{2}\frac{3\pi}{4}=\)

1. 1. 0
2. 2. \(\frac{1}{2}\)
3. 3. 1
4. 4. \(\frac{1}{3}\)

27. In a triangle ABC, \(\tan\frac{A}{2}\tan\frac{B}{2}+\tan\frac{B}{2}\tan\frac{C}{2}+\tan\frac{C}{2}\tan\frac{A}{2}=\)

1. 1. 0
2. 2. 1
3. 3. \(\frac{1}{2}\)
4. 4. \(\pi\)

28. If \(\delta\) is any angle, then \(\sin^{2}\delta\cos^{2}\delta=\)

1. 1. \(1-\cos2\delta\)
2. 2. \(1-\cos4\delta\)
3. 3. \(\frac{1}{4}(1-\cos4\delta)\)
4. 4. \(\frac{1}{8}(1-\cos4\delta)\)

29. \((4\cos^{2}9^{\circ}-3)(4\cos^{2}27^{\circ}-3)=\)

1. 1. \(\sin9^{\circ}\)
2. 2. \(\cos9^{\circ}\)
3. 3. \(\tan9^{\circ}\)
4. 4. \(\cot9^{\circ}\)

30. \(\cos^{4}\frac{\pi}{24}-\sin^{4}\frac{\pi}{24}=\)

1. 1. \(\frac{\sqrt{2}-\sqrt{3}}{2}\)
2. 2. \(\frac{\sqrt{2}+\sqrt{3}}{2}\)
3. 3. \(\frac{\sqrt{2}-\sqrt{6}}{4}\)
4. 4. \(\frac{\sqrt{2}+\sqrt{6}}{4}\)

31. If \(\tan A=\frac{2}{3}\), then \(\sin4A=\)

1. 1. \(\frac{8}{27}\)
2. 2. \(\frac{120}{169}\)
3. 3. \(\frac{144}{169}\)
4. 4. \(\frac{16}{27}\)

32. If \(|\sin\alpha-\cos\alpha|=\frac{3}{4}\), then \(|\sec2\alpha-\tan2\alpha|=\)

1. 1. \(\frac{12}{17}\)
2. 2. \(\frac{4}{\sqrt{23}}\)
3. 3. \(\frac{3}{\sqrt{23}}\)
4. 4. \(\frac{7}{\sqrt{23}}\)

33. If \(\theta\) does not lie in the second quadrant and \(\tan\theta=\frac{-3}{4}\), then \(\tan\frac{\theta}{2}+\sin2\theta=\)

1. 1. \(\frac{97}{75}\)
2. 2. \(-\frac{97}{75}\)
3. 3. \(-\frac{47}{75}\)
4. 4. \(\frac{47}{75}\)

34. \(\frac{1}{\sin250^{\circ}}+\frac{\sqrt{3}}{\cos290^{\circ}}=\)

1. 1. \(\frac{1}{\sqrt{3}}\)
2. 2. 4
3. 3. \(\frac{4}{\sqrt{3}}\)
4. 4. 1

35. If \(\sin\theta-\cos\theta=\frac{1}{\sqrt{3}}\), then \(\sin(2\theta)+\cos(4\theta)+\sin(6\theta)=\)

1. 1. \(\frac{37}{27}\)
2. 2. \(-\frac{37}{27}\)
3. 3. \(\frac{-43}{27}\)
4. 4. \(\frac{43}{27}\)

36. \(\sin^{4}\frac{\pi}{8}+\sin^{4}\frac{3\pi}{8}+\sin^{4}\frac{5\pi}{8}+\sin^{4}\frac{7\pi}{8}=\)

1. 1. \(\frac{1}{4}\)
2. 2. \(\frac{3}{8}\)
3. 3. \(\frac{3}{2}\)
4. 4. \(\frac{3}{4}\)

37. If two angles \(\alpha,\beta\) are such that \(0<\alpha,\beta<\frac{\pi}{4}\), \(\sqrt{1+\cos2\alpha}=\frac{3}{\sqrt{5}}\) and \(\sqrt{\frac{1-\cos2\beta}{1+\cos2\beta}}=\frac{1}{7}\), then \((2\alpha+\beta)=\)

1. 1. \(\frac{\pi}{3}\)
2. 2. \(\frac{\pi}{6}\)
3. 3. \(\frac{3\pi}{4}\)
4. 4. \(\frac{\pi}{4}\)

38. If \(\theta=\frac{\pi}{9}\), then \(1+27\tan^{2}\theta-33\tan^{4}\theta+\tan^{6}\theta=\)

1. 1. 3
2. 2. 4
3. 3. -3
4. 4. -11

39. \(\cot18^{\circ}\cot36^{\circ}+1=\)

1. 1. \(\sqrt{5+2\sqrt{5}}\)
2. 2. \(\sqrt{5-2\sqrt{5}}\)
3. 3. \(3-\sqrt{5}\)
4. 4. \(3+\sqrt{5}\)

40. \(\cos12^{\circ}\cos24^{\circ}\cos36^{\circ}\cos48^{\circ}\cos72^{\circ}\cos84^{\circ}=\)

1. 1. \(\frac{1}{32}\)
2. 2. \(\frac{1}{16}\)
3. 3. \(\frac{1}{64}\)
4. 4. \(\frac{1}{128}\)

41. If \(\sin\theta=\frac{3}{5}\) and \(\theta\) is not in the first quadrant, then \(15\sin2\theta-20\cos2\theta-7\tan2\theta=\)

1. 1. -4
2. 2. -12
3. 3. 12
4. 4. 4

42. \([1+\sec2\theta][1+\sec4\theta]=\)

1. 1. \(\tan\theta\tan4\theta\)
2. 2. \(4\cot\theta\tan4\theta\)
3. 3. \(\cot\theta\tan4\theta\)
4. 4. \(4\tan\theta\tan4\theta\)

43. \(\left(1+\cos\frac{\pi}{8}\right)\left(1+\cos\frac{2\pi}{8}\right)\left(1+\cos\frac{3\pi}{8}\right)\left(1+\cos\frac{4\pi}{8}\right)\ldots\left(1+\cos\frac{7\pi}{8}\right)=\)

1. 1. \(\frac{1}{16}\)
2. 2. \(\frac{1}{64}\)
3. 3. \(\frac{3}{16}\)
4. 4. \(\frac{3}{64}\)

44. If \(3\sin^{4}x+2\cos^{4}x=\frac{6}{5}\) and \(x\) is an acute angle, then \(\tan2x=\)

1. 1. \(\frac{2\sqrt{6}}{5}\)
2. 2. \(2\sqrt{6}\)
3. 3. \(\frac{3\sqrt{2}}{5}\)
4. 4. \(\frac{2\sqrt{3}}{5}\)

45. \(\cos\frac{\pi}{2^{2}}\cdot\cos\frac{\pi}{2^{3}}\cdot\cos\frac{\pi}{2^{4}}\ldots\cos\frac{\pi}{2^{10}}=\)

1. 1. \(\sin\left(\frac{\pi}{2^{10}}\right)\)
2. 2. \(\csc\left(\frac{\pi}{2^{10}}\right)\)
3. 3. \(\sin\left(\frac{\pi}{2^{10}}\right)\)
4. 4. \(\csc\left(\frac{\pi}{2^{10}}\right)\)

46. If \(\sin(\alpha+\beta)=5\sin(\alpha-\beta)\), then \(\frac{\sin2\beta}{5-\cos2\beta}=\)

1. 1. \(\tan(\alpha+\beta)\)
2. 2. \(\cot(\alpha+\beta)\)
3. 3. \(\cot(\alpha-\beta)\)
4. 4. \(\tan(\alpha-\beta)\)

47. If \(\cos A+\cos B+\cos C=0=\sin A+\sin B+\sin C\), then \(\cos(A-B)=\)

1. 1. 0
2. 2. \(\frac{1}{2}\)
3. 3. \(\frac{2}{3}\)
4. 4. \(\frac{1}{2}\)

48. If \(\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}=\frac{\sin(8\pi/7)}{8\sin(\pi/7)}\), then \(\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}\sin\frac{7\pi}{14}\sin\frac{9\pi}{14}\sin\frac{11\pi}{14}\sin\frac{13\pi}{14}=\)

1. 1. \(\frac{1}{16}\)
2. 2. \(\frac{1}{32}\)
3. 3. \(\frac{1}{64}\)
4. 4. \(\frac{1}{128}\)

49. If \(f(\theta)=\cos^{3}\theta+\cos^{3}\left(\frac{2\pi}{3}+\theta\right)+\cos^{3}\left(\theta-\frac{2\pi}{3}\right)\), then \(f\left(\frac{\pi}{5}\right)=\)

1. 1. \(\frac{-3(\sqrt{5}-1)}{16}\)
2. 2. \(\frac{3\sqrt{10}-2\sqrt{5}}{8}\)
3. 3. \(\frac{3\sqrt{10}+2\sqrt{5}}{8}\)
4. 4. \(\frac{3(\sqrt{5}+1)}{16}\)

50. If \(540^{\circ}<\theta<630^{\circ}\) and \(\tan\theta=\frac{5}{12}\), then \(\frac{\cos\frac{\theta}{2}-5\sin\frac{\theta}{2}}{\sqrt{-(12\sec\theta+5\csc\theta)}}=\)

1. 1. -26
2. 2. 26
3. 3. 1
4. 4. -1

51. If \(\cos\theta=\frac{-3}{5}\) and \(\pi<\theta<\frac{3\pi}{2}\), then \(\tan\frac{\theta}{2}+\sin\frac{\theta}{2}+2\cos\frac{\theta}{2}=\)

1. 1. -1
2. 2. 1
3. 3. -2
4. 4. 2

52. If \(\sin2\theta\) and \(\cos2\theta\) are solutions of \(x^{2}+ax-c=0\), then

1. 1. \(a^{2}-2c-1=0\)
2. 2. \(a^{2}+2c-1=0\)
3. 3. \(a^{2}+2c+1=0\)
4. 4. \(a^{2}-2c+1=0\)

53. If \(\tan\alpha=\frac{-12}{5}\), \(\cot\beta=\frac{7}{24}\), \(\alpha\) does not belong to second quadrant and \(\beta\) does not belong to the first quadrant, then \(\sqrt{13}\sin\frac{\alpha}{2}+\cos\frac{\beta}{2}+\tan\frac{\alpha}{2}\cot\frac{\beta}{2}=\)

1. 1. 31/10
2. 2. 19/10
3. 3. 21/10
4. 4. -9/10

54. \(\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7}\cos\frac{\pi}{14}\cos\frac{3\pi}{14}\cos\frac{5\pi}{14}=\)

1. 1. \(\frac{1}{16}[\sin\frac{\pi}{7}+\sin\frac{2\pi}{7}+\sin\frac{3\pi}{7}]\)
2. 2. \(\frac{1}{8}[\sin\frac{2\pi}{7}+\sin\frac{3\pi}{7}-\sin\frac{\pi}{7}]\)
3. 3. \(\frac{1}{32}[\sin\frac{2\pi}{7}+\sin\frac{3\pi}{7}-\sin\frac{\pi}{7}]\)
4. 4. \(\frac{1}{32}[\sin\frac{\pi}{7}-\sin\frac{2\pi}{7}+\sin\frac{3\pi}{7}]\)

1. Let A, B and C be three angles of a triangle ABC such that \(\cos A+\cos B+\cos C=\frac{3}{2}\), then the triangle ABC is

1. 1. Equilateral
2. 2. Right angled
3. 3. Isosceles but not equilateral
4. 4. Scalene

2. If \(\frac{\cos(\theta_1+\theta_2)}{\cos(\theta_1-\theta_2)}+\frac{\cos(\theta_3-\theta_4)}{\cos(\theta_3+\theta_4)}=0\), then \(\cot\theta_1\cot\theta_2\cot\theta_3\cot\theta_4=\)

1. 1. 1
2. 2. -1
3. 3. 2
4. 4. \(\frac{1}{2}\)

3. If \(\sin2\theta+\sin2\phi=\frac{1}{2}\) and \(\cos2\theta+\cos2\phi=\frac{3}{2}\), then \(\cos^{2}(\theta-\phi)=\)

1. 1. \(\frac{3}{8}\)
2. 2. \(\frac{5}{8}\)
3. 3. \(\frac{3}{4}\)
4. 4. \(\frac{5}{4}\)

4. If \(A+B+C=60^{\circ}\), then \(\cos(30^{\circ}-A)+\cos(30^{\circ}-B)+\cos(30^{\circ}-C)+\sin(A+B+C)=\)

1. 1. \(4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\)
2. 2. \(4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\)
3. 3. \(4\cos\frac{A}{2}\cos\frac{B}{2}\sin\frac{C}{2}\)
4. 4. \(4\cos\frac{A}{2}\sin\frac{B}{2}\cos\frac{C}{2}\)

5. If \(\cos\left(\frac{\alpha-\beta}{2}\right)=2\cos\left(\frac{\alpha+\beta}{2}\right)\), then \(\tan\frac{\alpha}{2}\tan\frac{\beta}{2}=\)

1. 1. \(\frac{1}{2}\)
2. 2. \(\frac{1}{4}\)
3. 3. \(\frac{1}{3}\)
4. 4. \(\frac{1}{8}\)

6. If \(\sin\alpha-\cos\alpha=m\) and \(\sin2\alpha=n-m^{2}\), where \(-\sqrt{2}\leq m\leq\sqrt{2}\), then 'n' is equal to

1. 1. 0
2. 2. 1
3. 3. 2
4. 4. -2

7. If \(A+B+C=\frac{3\pi}{2}\), then \(\cos2A+\cos2B+\cos2C=\)

1. 1. \(1-4\sin A\sin B\sin C\)
2. 2. \(1+4\sin A\sin B\sin C\)
3. 3. \(1-2\sin A\sin B\sin C\)
4. 4. \(1+2\sin A\sin B\sin C\)

8. \(\cos\frac{7\pi}{8}+\cos\frac{\pi}{4}+\cos\left(-\frac{\pi}{8}\right)-1=\)

1. 1. \(4\cos\frac{\pi}{16}\cos\frac{3\pi}{4}\cos\frac{5\pi}{8}\)
2. 2. \(4\cos\frac{\pi}{16}\cos\frac{\pi}{4}\sin\frac{5\pi}{8}\)
3. 3. \(4\cos\frac{\pi}{16}\cos\frac{3\pi}{8}\cos\frac{9\pi}{16}\)
4. 4. \(-4\cos\frac{\pi}{16}\cos\frac{5\pi}{8}\cos\frac{\pi}{16}\)

9. If \(A+B+C=45^{\circ}\), then \(\cos(2S-A)+\cos(2S-B)-\cos(2S-C)-\cos2S=\)

1. 1. \(4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\)
2. 2. \(4\cos\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\)
3. 3. \(4\sin\frac{A}{2}\cos\frac{B}{2}\sin\frac{C}{2}\)
4. 4. \(4\sin\frac{A}{2}\sin\frac{B}{2}\cos\frac{C}{2}\)

10. If \(\frac{\sin(x+y)}{\sin(x-y)}=\frac{a+b}{a-b}\), then \(\frac{\tan x}{\tan y}=\)

1. 1. \(\frac{b}{a}\)
2. 2. \(\frac{a}{b}\)
3. 3. \(ab\)
4. 4. \(a^{b}\)

11. If \(x\neq-y\) and \(\sin x+\sin y=3(\cos y-\cos x)\), then \(\tan(x-y)=\)

1. 1. \(\frac{\sqrt{3}}{2}\)
2. 2. -1
3. 3. \(\frac{3}{4}\)
4. 4. 1

12. In a \(\Delta ABC\), if \(\cos A+\cos B+\cos C=a+b\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\), then \(a+b=\)

1. 1. 3
2. 2. 0
3. 3. 1
4. 4. 5

13. If \(\tan\beta=\frac{\tan\alpha+\tan\gamma}{1+\tan\alpha\tan\gamma}\), then \(\frac{\sin2\alpha+\sin2\gamma}{1+\sin2\alpha\sin2\gamma}=\)

1. 1. \(\sin2\beta\)
2. 2. \(\cos2\beta\)
3. 3. \(\tan2\beta\)
4. 4. \(\sec2\beta\)

14. The value of \(\frac{\sin\theta+\sin3\theta}{\cos\theta+\cos3\theta}\) is

1. 1. \(\cos2\theta\)
2. 2. \(\cot2\theta\)
3. 3. \(\tan2\theta\)
4. 4. \(\csc\theta+\sin\theta\)

15. Let \(\alpha,\beta\) be two real numbers such that \(\pi<(\alpha-\beta)<3\pi\). If \(\sin\alpha+\sin\beta=\frac{-21}{65}\) and \(\cos\alpha+\cos\beta=\frac{27}{65}\), then \(\cos\left(\frac{\beta-\alpha}{2}\right)=\)

1. 1. \(\frac{3}{\sqrt{130}}\)
2. 2. \(-\frac{3}{\sqrt{130}}\)
3. 3. \(\frac{130}{\sqrt{3}}\)
4. 4. \(-\frac{\sqrt{130}}{3}\)

16. Let \(x,y,z\) be real numbers and \(x\geq y\geq z\geq\frac{\pi}{12}\). If \(x+y+z=\frac{\pi}{2}\), then the minimum value of \(\cos x\cdot\sin y\cdot\cos z\) is

1. 1. \(\frac{1}{2}\)
2. 2. \(\frac{1}{4}\)
3. 3. \(\frac{1}{6}\)
4. 4. \(\frac{1}{8}\)

17. If \(\sin\left(x+\frac{\pi}{3}\right)+\sin\left(x-\frac{\pi}{3}\right)=1\), then the value of \(x\) in the interval \([0,\pi]\) is

1. 1. \(\frac{\pi}{2}\)
2. 2. \(\frac{\pi}{3}\)
3. 3. 0
4. 4. \(\frac{\pi}{4}\)

18. In a triangle ABC, \(\sin2A+\sin2B+\sin2C=\)

1. 1. \(4\sin A\sin B\sin C\)
2. 2. \(2\sin A\sin B\sin C\)
3. 3. \(4\cos A\cos B\cos C\)
4. 4. \(2\sin A\cos B\cos C\)

19. \(\frac{\sqrt{2}\cos45^{\circ}+\cos56^{\circ}+\cos58^{\circ}-\cos66^{\circ}}{\sqrt{2}\cos28^{\circ}\cos29^{\circ}\sin33^{\circ}}=\)

1. 1. \(\sqrt{2}\)
2. 2. \(2\sqrt{2}\)
3. 3. \(\frac{\sqrt{2}}{2}\)
4. 4. \(4\sqrt{2}\)

20. If \(A+B+C=\frac{3\pi}{2}\), then \(4\sin A\sin B\sin C+\cos2A+\cos2B+\cos2C=\)

1. 1. \(-\sin(A+B+C)\)
2. 2. \(\cos(A+B+C)\)
3. 3. \(\sin(A+B+C)\)
4. 4. \(2-\cos(A+B+C)\)

21. \(\cos^{2}76^{\circ}+\sin^{2}46^{\circ}+\sin76^{\circ}\cos46^{\circ}=\)

1. 1. \(\frac{3}{4}\)
2. 2. 1
3. 3. \(\frac{5}{4}\)
4. 4. 2

22. If \(A+B+C=\frac{\pi}{2}\), then \(\sqrt{2}\cos\left(\frac{\pi}{4}-A\right)+\sqrt{2}\cos\left(\frac{\pi}{4}-B\right)+\sqrt{2}\cos\left(\frac{\pi}{4}-C\right)+1=\)

1. 1. \(4\sqrt{2}\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\)
2. 2. \(4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\)
3. 3. \(4\sin\frac{A}{2}\sin\frac{B}{2}\cos\frac{C}{2}\)
4. 4. \(4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\)

23. If \(a\tan\alpha+b\tan\beta=(a+b)\tan\left(\frac{\alpha+\beta}{2}\right)\) and \(\alpha-\beta\neq2n\pi\), then \(\cos\beta=\)

1. 1. \(\frac{a}{b}\)
2. 2. \(\frac{a+b}{a-b}\)
3. 3. \(\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\)
4. 4. \(\frac{b}{a}\)

24. If \(\cos^{3}x\sin4x=\sum_{r=0}^{n}a_{r}\sin rx\ \forall x\in\mathbb{R}\), then \(a_{3}+a_{5}:a_{1}+a_{7}=\)

1. 1. 1:3
2. 2. 1:1
3. 3. 2:1
4. 4. 3:1

25. In \(\Delta ABC\), \(\frac{\sin2A+\sin2B+\sin2C}{\cos A+\cos B+\cos C-1}=\)

1. 1. \(2[\sin A+\sin B+\sin C]\)
2. 2. \(\sin A+\sin B+\sin C\)
3. 3. \(4[\sin A+\sin B+\sin C]\)
4. 4. \(8[\sin A+\sin B+\sin C]\)

26. \(\cot16^{\circ}\cot44^{\circ}+\cot44^{\circ}\cot76^{\circ}-\cot76^{\circ}\cot16^{\circ}=\)

1. 1. 1
2. 2. -1
3. 3. -3
4. 4. 3

27. In \(\Delta ABC\), if \(\cos^{2}A+\cos^{2}B+\cos^{2}C=1\), then \(\Delta ABC\) is

1. 1. Equilateral
2. 2. Isosceles
3. 3. Right angled
4. 4. Scalene

28. If two acute angles A and B are such that \(A\neq B\) and \(\frac{x}{y}=\frac{\cos A}{\cos B}\), then \(\frac{x\tan A-y\tan B}{x+y}=\)

1. 1. \(\tan\left(\frac{A-B}{2}\right)\)
2. 2. \(\tan\left(\frac{B-A}{2}\right)\)
3. 3. \(\tan\left(\frac{A+B}{2}\right)\)
4. 4. \(\cot\left(\frac{A+B}{2}\right)\)

29. If \(m\tan(\theta-30^{\circ})=n\tan(\theta+120^{\circ})\), then \(\frac{m+n}{m-n}=\)

1. 1. \(2\cos2\theta\)
2. 2. \(2\cos^{2}\theta\)
3. 3. \(\tan2\theta\)
4. 4. \(2\sin2\theta\)

30. If \(\cos\alpha+\cos\beta=\frac{24}{25}\) and \(\sin\alpha+\sin\beta=\frac{7}{25}\), then \(\cos(\alpha+\beta)=\)

1. 1. \(\frac{24}{25}\)
2. 2. \(\frac{7}{25}\)
3. 3. \(\frac{13}{25}\)
4. 4. \(\frac{12}{25}\)

31. If \(A+B+C+D=2\pi\), then \(\cos A-\cos B+\cos C-\cos D=\)

1. 1. \(-4\sin\frac{A+B}{2}\cos\frac{A+C}{2}\sin\frac{A+D}{2}\)
2. 2. \(4\sin\frac{A+B}{2}\sin\frac{A+C}{2}\sin\frac{A+D}{2}\)
3. 3. \(4\cos\frac{A+B}{2}\cos\frac{A+C}{2}\cos\frac{A+D}{2}\)
4. 4. \(4\sin\frac{A+B}{2}\cos\frac{A+C}{2}\sin\frac{A+D}{2}\)

32. \(\sin6^{\circ}+\sin54^{\circ}+\sin126^{\circ}+\cos156^{\circ}=\)

1. 1. \(\frac{\sqrt{5}+1}{4}\)
2. 2. \(\frac{\sqrt{5}-1}{4}\)
3. 3. \(-\frac{1}{2}\)
4. 4. \(\frac{3}{4}\)

1. If \(\cos(x)+\cos^{2}(x)=1\), then \(\sin^{2}(x)+\sin^{4}(x)\) is equal to

1. 1. 0
2. 2. 1
3. 3. -1
4. 4. 2

2. If \(\theta\) lies in the third quadrant and \(\cos\theta=\frac{-3}{5}\), find value of \(\tan\theta\).

1. 1. \(\frac{2}{3}\)
2. 2. \(-\frac{2}{3}\)
3. 3. \(-\frac{4}{3}\)
4. 4. \(\frac{4}{3}\)

3. Find the value of \(\csc750^{\circ}-2\cot765^{\circ}\)

1. 1. 0
2. 2. 1
3. 3. 2
4. 4. -1

4. Let \(f(x)=\cos(ax)+\sin(x)\) be periodic, then a must be

1. 1. Irrational
2. 2. Rational
3. 3. Positive real number
4. 4. Negative real number

5. The minimum and maximum values of \(\cos\left(x+\frac{\pi}{3}\right)+2\sqrt{2}\sin\left(x+\frac{\pi}{3}\right)\) are respectively

1. 1. \((-2\sqrt{3}-1)\) & \(2\sqrt{3}-1\)
2. 2. \((-1+2\sqrt{2})\) & \(2\sqrt{2}+1\)
3. 3. -3 and 3
4. 4. -2 and 2

6. Let a be maximum value of \((3\cos\theta-4\sin\theta)\) and \(\theta\neq\frac{n\pi}{2}\). If \(\alpha=a\sin^{2}\theta\cos^{3}\theta\) and \(\beta=a\sin^{3}\theta\cos^{2}\theta\), then \(\sqrt{\frac{(\alpha^{2}+\beta^{2})^{5}}{(\alpha\beta)^{4}}}=\)

1. 1. \(5\sin\frac{\theta}{2}\cos^{2}\frac{\theta}{2}\)
2. 2. \(-3\sin\theta\)
3. 3. 5
4. 4. 16

7. The period of \(\frac{\sin x}{\cos3x}+\frac{\sin3x}{\cos9x}+\frac{\sin9x}{\cos27x}+\frac{\sin27x}{\cos81x}\) is

1. 1. \(\frac{2\pi}{3}\)
2. 2. \(\frac{\pi}{81}\)
3. 3. \(\frac{2\pi}{4}\)
4. 4. \(\frac{\pi}{4}\)

8. Match the items of List-I with those of List-II.

1. 1. \(A\to III, B\to V, C\to II, D\to IV\)
2. 2. \(A\to III, B\to I, C\to II, D\to IV\)
3. 3. \(A\to I, B\to III, C\to IV, D\to V\)
4. 4. \(A\to II, B\to I, C\to III, D\to V\)

9. The period of \(\cos(3x+5)+7\) is

1. 1. \(\frac{2\pi}{5}\)
2. 2. \(\frac{2\pi}{3}\)
3. 3. \(\frac{2\pi}{15}\)
4. 4. \(\frac{2\pi}{7}\)

10. Minimum value of \(5\tan^{2}\alpha+\frac{9}{\tan^{2}\alpha}+4\sec^{2}\alpha\) is

1. 1. 24
2. 2. 22
3. 3. 32
4. 4. 28

11. The larger of \(\cos(\log\theta)\) and \(\log(\cos\theta)\) if \(e^{-\pi/2}<\theta<\pi/2\) is

1. 1. \(\cos(\log\theta)\)
2. 2. \(\log(\cos\theta)\)
3. 3. None of function is larger
4. 4. One of the two function is undefined on domain even to compare

12. Let \(y=4\sin^{2}\theta-\cos2\theta\). If \(l\) and \(m\) are the minimum and maximum values of y respectively, then

1. 1. \(lm=\frac{m}{l}\)
2. 2. \(lm=\frac{l}{m}\)
3. 3. \(l+m=\frac{l}{m}\)
4. 4. \(\frac{lm}{l-m}=1+m\)

13. The period of \(\tan ky+\sin ky\), where \(k=1+4+9+\ldots 20\) terms, is

1. 1. \(\frac{\pi}{1435}\)
2. 2. \(\frac{2\pi}{1435}\)
3. 3. \(\pi\)
4. 4. \(2\pi\)

14. Let \(\alpha\) be the period of \(3\sin\frac{\pi x}{3}-\cos\frac{\pi x}{2}+\tan\frac{\pi x}{4}\), \(\beta\) be the period of \(\sin^{2}\left(\frac{\pi}{7}+\frac{x}{4}\right)-\sin^{2}\left(\frac{\pi}{7}-\frac{x}{4}\right)\) and \(\gamma\) be the period of \(\cos^{4}x+\sin^{4}x\). Then \(\frac{\alpha\gamma}{\beta}=\)

1. 1. \(\frac{3}{2}\)
2. 2. \(\frac{3}{4}\)
3. 3. 3
4. 4. 6

15. The range of \(\frac{1}{\sin^{2}x+3\sin x\cos x+5\cos^{2}x}\) is

1. 1. \(\left[2,\frac{11}{2}\right]\)
2. 2. \(\left[\frac{1}{2},\frac{11}{2}\right]\)
3. 3. \(\left[\frac{2}{11},\frac{1}{2}\right]\)
4. 4. \(\left[\frac{2}{11},2\right]\)

16. Match the ranges of the functions given in List-A with those of the items given in List-B.

1. 1. (I)→(c) (II)→(a) (III)→(d) (IV)→(b)
2. 2. (I)→(c) (II)→(d) (III)→(a) (IV)→(b)
3. 3. (I)→(b) (II)→(d) (III)→(a) (IV)→(e)
4. 4. (I)→(b) (II)→(e) (III)→(d) (IV)→(c)

17. The period of the function \(f(x)=e^{\log(\sin x)}+(\tan x)^{3}-\csc(3x-5)\) is

1. 1. \(\pi\)
2. 2. \(\pi/2\)
3. 3. \(2\pi\)
4. 4. \(\frac{2\pi}{3}\)