1. If sinA+sinB=2, then what is the value of cosA-cosB (a) 0 (b) 1 (c) 2 (d) 3

2. In triangle ABC , what is the value of sin⁡\(\cfrac{(B+C)}{2} \) ? (a) sin⁡\(\cfrac{A}{2}\) (b) sinA (c) cosA (d) cos⁡ \(\cfrac{A}{2}\)

3. If \( 2 \cos \theta = 1 \), what is the value of \( \theta \) ? (a) 10° (b) 15° (c) 60° (d) 30°

4. If sin 51° = \(\cfrac{a}{\sqrt{a^2 + b^2}}\), then what is the value of tan 51° + tan 39°? (a) \(\cfrac{a^2-b^2}{ab}\) (b) \(\cfrac{a^2+b^2}{2ab}\) (c) \(\cfrac{a^2+b^2}{ab}\) (d) \(\cfrac{a^2-b^2}{2ab}\)

5. If \(\sin \theta + \cos \theta = \sqrt{2}\), then what is the value of \(\theta\)? (a) \(\cfrac{\pi}{2}\) (b) \(\cfrac{\pi}{3}\) (c) \(\pi\) (d) \(\cfrac{\pi}{4}\)

6. If, tanθ=cot3θ, then the value of sin2θ be: True / False

7. Find the value of: (sin43°cos47°+cos43°sin47°) (a) 0 (b) 1 (c) sin4° (d) cos4°

8. The value of (sin43°cos47° +cos43°sin47°) is: (a) 0 (b) 1 (c) sin4° (d) cos4°

9. If \( \tan\left(\cfrac{\pi}{2} - \cfrac{\alpha}{2}\right) = \sqrt{3} \), then what is the value of \( \cos\alpha \)? (a) \(\cfrac{1}{2}\) (b) \(\cfrac{\sqrt3}{2}\) (c) \(\cfrac{1}{\sqrt2}\) (d) 1

10. The measure of a supplementary angle of a certain angle is four times the measure of its complementary angle. What is the measure of that angle? (a) 30° (b) 60° (c) 45° (d) None of the above

11. If A + B = 90° and tanA = \(\cfrac{3}{4}\), then the value of cotB is - (a) \(\cfrac{3}{4}\) (b) \(\cfrac{4}{3}\) (c) \(\cfrac{3}{5}\) (d) \(\cfrac{5}{3}\)

12. In triangle ABC, sin\(\cfrac{(B+C)}{2}\) = (a) sin⁡\(\cfrac{A}{2}\) (b) cos⁡\(\cfrac{A}{2}\) (c) sinA (d) cosA

13. What is the value of \( \tan 4^\circ \cdot \tan 43^\circ \cdot \tan 47^\circ \cdot \tan 86^\circ \)? (a) -1 (b) 1 (c) 0 (d) -2

14. ABC is a triangle. Find the value of \(\sin\left(\cfrac{B+C}{2}\right)\) (a) sin⁡\(\cfrac{A}{2}\) (b) cos⁡\(\cfrac{A}{2}\) (c) sinA (d) cosA

15. If \( \tan A \tan B = 1\), then the value of \( \tan \cfrac{(A+B)}{2} \) will be – (a) 1 (b) √3 (c) \(\cfrac{1}{√3}\) (d) None of these

16. If the complement of a \(3x^\circ\) angle is \(6x^\circ\), then what is the value of \(x\)? (a) 10 (b) 9 (c) 20 (d) 12

17. If \(\theta\) is a positive acute angle and \( \sin \theta - \cos \theta = 0 \), then the value of \(\cot 2\theta\) is – True / False

18. Given: \[ \theta + \phi = \frac{7\pi}{12},\quad \tan\theta = \sqrt{3} \Rightarrow \theta = \frac{\pi}{3} \] \[ \phi = \frac{7\pi}{12} - \frac{\pi}{3} = \frac{7\pi}{12} - \frac{4\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4} \] \[ \tan\phi = \tan\left(\frac{\pi}{4}\right) = 1 \] (a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{\sqrt3}{2}\)

19. What is the value of \( \sin 12^\circ \cdot \cos 18^\circ \cdot \sec 78^\circ \cdot \csc 72^\circ \)? (a) 1 (b) \(\cfrac{1}{2}\) (c) \(\cfrac{1}{\sqrt2}\) (d) \(\cfrac{\sqrt3}{2}\)

20. If \( \tan \theta \cos 60° = \cfrac{√3}{2} \), find the value of \(\sin(\theta - 15°)\) True / False

21. If \(\sin(3x - 20^\circ) = \cos(3y + 20^\circ)\), then what is the value of \(x + y\)? (a) 60° (b) 30° (c) 45° (d) 90°

22. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)? (a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

23. If sin(45° + θ) = cos(2θ - 30°), then θ = ? (a) 25° (b) 45° (c) 15° (d) 35°

24. "The value of sin ⁡ 2 22 ∘ + sin ⁡ 2 68 ∘ will be (a) 0 (b) 1 (c) -1 (d) 2

25. From the equation \( \sin(90^\circ + \theta) = \cos(120^\circ - 3\theta) \), the value of \( \theta \) is — (a) 30° (b) 60° (c) 45° (d) None of the above

26. If \( \sec\theta = \csc 5\theta \), then the value of \( \theta \) is — (a) 5° (b) 10° (c) 15° (d) 30°

27. If \( \csc\theta \tan\theta = 2 \), then what is the value of \( \theta \)? (a) 60° (b) 45° (c) 30° (d) 90°

28. What is the value of \( \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ \)? (a) -1 (b) 1 (c) 0 (d) -2

29. What is the value of \( \sec^2 12^\circ - \frac{1}{\tan^2 78^\circ} \)? (a) 0 (b) 1 (c) -1 (d) 2

30. From the equation \( \sin(90^\circ + \theta) = \cos(120^\circ - 3\theta) \), what is the value of \( \theta \)? (a) 30° (b) 60° (c) 45° (d) None of the above

31. If \(sin θ = cos θ\), then the value of \(2θ\) is—? (a) 30° (b) 60° (c) 45° (d) 90°

32. If \(sinθ−cosθ=0,\) \( (0°<θ<90°)\) and \(secθ+cosecθ=x\), then the value of \(x\) is—? (a) \(1\) (b) \(2\) (c) \(\sqrt2\) (d) \(2\sqrt2\)

33. \(\theta\) is a positive acute angle, and if \( \tan\theta = \cot\theta \), then what is the value of \(\theta\)? (a) 40° (b) 45° (c) 60° (d) 20°

34. What is the value of \( \tan 35^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 55^\circ \)? True / False

35. If \( \sin 51^\circ = \cfrac{a}{\sqrt{a^2 + b^2}} \), then what is the value of \( \tan 51^\circ + \tan 39^\circ \)? (a) \(\cfrac{a^2+b^2}{ab}\) (b) \(\cfrac{a^2-b^2}{ab}\) (c) \(\cfrac{a-b}{ab}\) (d) \(\cfrac{a+b}{ab}\)

36. The simplest value of \(sec^2 27° - cot^2 63°\) is 1. True / False

37. The value of cos 0° × cos 1° × cos 2° × … × cos 90° is 1. True / False

38. \(\sin^2 21^\circ + \cos^2 89^\circ = 1\). True / False

39. The values of \(\cos 36^\circ\) and \(\sin 54^\circ\) are equal. True / False

40. If \(0^\circ < \theta < 90^\circ\), then \( \cos \theta > \cos^2 \theta \). True / False

41. If sin A + sin 2B + sin 3C = 3, then what is the value of (A + 2B + 3C)?

42. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?

43. If \( \sin(\theta - 30^\circ) = \frac{1}{2} \), then the value of \( \cos \theta \) is _____.

44. If \(\sin(A + B) = 1\) and \(\cos(A - B) = 1\), then find the value of \(\cot 2A\), given that \(0^\circ \leq (A + B) \leq 90^\circ\) and \(A \geq B\).

45. If \(\angle A + \angle B = 90^\circ\), then prove that \[ 1 + \cfrac{\tan A}{\tan B} = \tan^2 A \sec^2 B \]

46. If \( \tan(θ + 15^\circ) = \sqrt{3} \), then find the value of \( \sinθ + \cosθ \).

47. What is the value of \((\cos^2 20^\circ + \cos^2 70^\circ)\)?

48. If \(\sin (\theta + 30^\circ) = \cos 15^\circ\), then what is the value of \(\cos 2\theta\)?

49. If \(tan 35° \cdot tan 55° = \sin θ\), then the smallest positive value of \(θ\) will be ——.

50. If \(\tan \alpha = \cot \beta\), find the value of \(\cos(\alpha + \beta)\), where \(0^\circ < \alpha, \beta < 90^\circ\).

51. In triangle \(\triangle ABC\), prove that: \[ \sin\left(\frac{A + B}{2}\right) + \cos\left(\frac{B + C}{2}\right) = \cos\left(\frac{C}{2}\right) + \sin\left(\frac{A}{2}\right) \]

52. Write the simplest value of \(\tan 70^\circ \times \tan 20^\circ\).

53. Find the value of: \(\cfrac{\sec 17^\circ}{\csc 73^\circ} + \cfrac{\tan 68^\circ}{\cot 22^\circ} + \cos^2 44^\circ + \cos^2 46^\circ\)

54. The simplest value of \(\cfrac{\cos 53^\circ}{\sin 37^\circ}\) is _____.

55. \(\cot 12^\circ \cdot \cot 38^\circ \cdot \cot 52^\circ \cdot \cot 78^\circ \cdot \cot 60^\circ = \frac{1}{\sqrt{3}}\)

56. Find the value of the complement of 67°.

57. If \( \sin10θ = \cos8θ \) and \(10θ\) is a positive acute angle, find the value of \( \tan9θ \).

58. If \(\tan \theta \cos 60° = \cfrac{\sqrt{3}}{2}\), then the value of \(\sin (\theta - 15°)\) will be _____ .

59. If \(\angle A\) and \(\angle B\) are complementary angles, then \(\angle A + \angle B =\) _____.

60. If \(∠P + ∠Q = 90^\circ\), then prove that \[ \sqrt{\frac{\sin P}{\cos Q}} - \sin P \cos Q = \cos^2 P \]

61. If \(\tan(x + 15^\circ) = 1\), then \(x + 15^\circ = 45^\circ\) So, \(x = 30^\circ\) Therefore, \(\tan x = \tan 30^\circ = \frac{1}{\sqrt{3}}\)

62. If \(sin17°=\cfrac{x}{y}\), then show that \(sec17°-sin73°= \cfrac{x^2}{y\sqrt{y^2-x^2}}\).

63. If \( \tan 2A = \cot (A - 30^\circ) \), then find the value of \( \sec (A + 20^\circ) \).

64. Two angles are called complementary if their sum is _____ degrees.

65. Show that: \(\csc^2 22^\circ \cdot \cot^2 68^\circ = \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 68^\circ\)

66. Find the simplest value of \(\tan\cfrac{3\pi}{20} \cdot \tan\cfrac{4\pi}{20} \cdot \tan\cfrac{5\pi}{20} \cdot \tan\cfrac{6\pi}{20} \cdot \tan\cfrac{7\pi}{20}\).

67. If \(\sec\theta = \csc\phi\), then \(\csc(\theta + \phi) = \sqrt{2}\).

68. If \( \tan 4θ \tan 6θ = 1 \) and \(6θ\) is a positive acute angle, find the value of \(θ\).

69. Prove that \[ \frac{\tan 47^\circ + \cot 27^\circ}{\tan 43^\circ + \cot 63^\circ} = \tan 47^\circ \cdot \cot 27^\circ \]

70. \[ \cos 10^\circ \cos 20^\circ \cos 60^\circ \cos 70^\circ \cos 80^\circ \cos 90^\circ = _____ \]

71. \(\sin(2x-20°)=\cos(2y+20°)\)
বা, \(\sin(2x-20°)=\sin[90^o-(2y+20°)]\)
বা, \(2x-20°= 90^o-(2y+20°)\)
বা, \(2x-20°= 90^o-2y-20°\)
বা, \(2x+2y= 90^o-20°+20°\)
বা, \(2(x+y)= 90^o\)
বা, \((x+y)= 45^o\)

\(\therefore \tan(x+y)=\tan 45^o=1\) (Answer)
- translate in english

72. If \(x = a\cos(90^\circ - \theta)\), \(y = b\cot(90^\circ - \theta)\), then prove that \(\cfrac{a^2}{x^2} - \cfrac{b^2}{y^2} = 1\).

73. If \(\tan(\theta + 15^\circ) = \sqrt{3}\), then what is the value of \(\sin \theta\)?

74. If \(\tan 2\theta \cdot \tan 3\theta = 1\), then find the value of \(\theta\), given that \(0 \leq \theta \leq \cfrac{\pi}{2}\).

75. If \( \cot 67\frac{1^\circ}{2} = x (>0) \), determine the value of \( \sin 22\frac{1^\circ}{2} \).

76. If \(\cos 52^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 38^\circ\)?

77. If sec 3θ = cosec 2θ and 0° < 5θ < 90°, then what is the value of θ?

78. If \( \sin 23^\circ = p \), then express the value of \( \sin 67^\circ \) in terms of \( p \).

79. Show that: \[ \sec^2 13^\circ - \cot^2 77^\circ = 1 \]

80. If \[ \tan \theta \cdot \cos 60^\circ = \frac{\sqrt{3}}{2} \] then find the value of \[ \sin(\theta - 15^\circ) \] given that \(0^\circ < \theta < 90^\circ\).

81. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

82. If \(\angle A + \angle B = 90^\circ\), then prove that \(1 + \frac{\tan A}{\tan B} = \csc^2 B\).

83. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

84. Given that \(A + B = 90^\circ\), prove that \[ \tan A + \tan B = \frac{\csc^2 B}{\sqrt{\csc^2 B - 1}} \]

85. In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

86. What is the value of \( \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \ldots \times \tan 89^\circ \)?

87. If \(\sin(2x + y) = \cos(4x - y)\), find the value of \(\tan 3x\).

88. If \(\cos 43^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 47^\circ\)?