1. If the measure of an angle in degrees is \(x\) and in radians is \(y\), then what is the value of \(\cfrac{x}{y}\)? (a) \(\cfrac{π}{180}\) (b) \(\cfrac{180}{π}\) (c) \(\cfrac{π}{2}\) (d) \(\cfrac{2}{π}\)

2. If the three sides of a triangle are \[ \sec\theta,\ 1,\ \tan\theta \quad (\theta \ne 90^\circ) \] then, what is the measure of the largest angle of the triangle? (a) 30° (b) 45° (c) 60° (d) 90°

3. Minimum value of \( \tan \theta + \cot \theta \) (a) 0 (b) 2 (c) -2 (d) 1

4. If \( \sin\theta \cos\theta = \frac{1}{2} \), then what is the value of \( (\sin\theta + \cos\theta)^2 \)? (a) 1 (b) 3 (c) 2 (d) 4

5. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)? (a) 1 (b) 1 (c) 0 (d) None of the above

6. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)? (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1

7. What is the value of \(\cfrac{\sin \theta}{\csc \theta} + \cfrac{\cos \theta}{\sec \theta} - 2\)? (a) 0 (b) -1 (c) 4 (d) None of the above

8. If \( a\cos\theta + b\sin\theta = c \), then what is the value of \( a\sin\theta - b\cos\theta \)? (a) \(\pm\sqrt{a^2-b^2+c^2}\) (b) \(\pm\sqrt{a^2+b^2-c^2}\) (c) \(\pm\sqrt{a^2-b^2-c^2}\) (d) \(\pm\sqrt{b^2+c^2-a^2}\)

9. If \( \tan\alpha + \cot\alpha = \sqrt{3} \), then what is the value of \( \tan^3\alpha + \cot^3\alpha \)? (a) \(2\sqrt3\) (b) (c) (d)

10. If \( \cot\alpha = \tan(\beta + \gamma) \), then what is the value of \( \sin(\alpha + \beta + \gamma) \)? (a) 1 (b) 2 (c) 4 (d) \(\cfrac{3}{4}\)

11. If \( \sec\theta = m \) and \( \tan\theta = n \), then which of the following (a) \(m=n\) (b) \(m\gt n\) (c) \(m\lt n\) (d) None of the above

12. If \(2\sin 2\theta - \sqrt{3} = 0\), then what is the value of \(\csc \theta\)? (a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{2}{\sqrt3}\) (d) 2

13. The simplest value of \(\sin\theta \times \tan\theta + \cos\theta\) is — (a) \(cos\theta\) (b) \(tan\theta\) (c) \(cosec\theta\) (d) \(sec\theta\)

14. If \(\cos^2\theta - \sin^2\theta = \frac{1}{2}\), then the value of \(\tan\theta\) is — (a) \(-\cfrac{1}{\sqrt3} (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{2}{3}\)

15. The value of \(\cfrac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1}\) is – (a) \(\cfrac{1+cos\theta}{sin\theta}\) (b) \(\cfrac{1+sin\theta}{cos\theta}\) (c) \(\cfrac{1+tan\theta}{sec\theta}\) (d) \(\cfrac{1+sec\theta}{sin\theta}\)

16. What will be the minimum value of 4 cos ⁡ ???? − 3 sin ⁡ ???? ? (a) 1 (b) 2 (c) 0 (d) None of the above

17. If \( \sin\theta - \cos\theta = \frac{7}{13} \), then the value of \( \sin\theta + \cos\theta \) is — (a) \(\cfrac{13}{17}\) (b) \(\cfrac{17}{13}\) (c) \(\cfrac{13}{7}\) (d) None of the above

18. What is the value of \(\cfrac{9}{\csc^2\theta} + 4\cos^2\theta + \cfrac{5}{1 + \tan^2\theta}\)? (a) 1 (b) 0 (c) 3 (d) 9

19. যদি \(cot\alpha=tan(\beta+\gamma) \) হয়, তবে \(sin(\alpha+\beta+\gamma)\)= কত ? - translate in english (a) 1 (b) \(\cfrac{\sqrt3}{2}\) (c) 0 (d) \(\cfrac{1}{2}\)

20. If \( \csc^2 A = 4 - \sec^2 A \), then what is the value of \( A \)? (a) \(\cfrac{\pi}{6}\) (b) \(\cfrac{\pi}{3}\) (c) \(\cfrac{\pi}{2}\) (d) \(\cfrac{\pi}{4}\)

21. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)? (a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

22. If \(3x = \csc \alpha\) and \(y = \cot \alpha\), then the value of \(3\left(x^2 - \cfrac{1}{x^2}\right)\) is - (a) (b) (c) (d)

23. What is the value of \(\cfrac{1}{\tan\theta+\cfrac{1}{\tan\theta}}\)? (a) \(tan\theta\) (b) \(sin 2\theta\) (c) \(cos \theta\) (d) \(sin\theta cos\theta\)

24. If \( \tan\theta + \cot\theta = 2 \), then the value of \( \theta \) will be — (a) \(\cfrac{\pi}{2}\) (b) \(\cfrac{\pi}{4}\) (c) \(\pi\) (d) \(\cfrac{\pi}{6}\)

25. If \(2y \cos \theta = x \sin \theta\) and \(2x \sec \theta - y \cosec \theta = 3\), then \(x^2 + 4y^2 = ?\) (a) 2 (b) 1 (c) 0 (d) None of the above

26. What is the value of \[ \frac{2\tan 30^\circ}{1 - \tan^2 30^\circ}? \] (a) \(\cfrac{1}{\sqrt3}\) (b) \(\cfrac{2}{\sqrt3}\) (c) \(2\sqrt3\) (d) \(\sqrt3\)

27. If \( \sin^2A + \sin^4A = 1 \), then what is the value of \( \tan^2A - \tan^4A \)? (a) 1 (b) 0 (c) -1 (d) 2

28. If \( \cos\theta - \sin\theta = \sqrt{2} \sin\theta \), then what is the value of \( \cos\theta + \sin\theta \)? (a) \(2 cos\theta\) (b) \(\sqrt2 sin \theta\) (c) \(2 sin\theta\) (d) \(\sqrt2 cos \theta\)

29. If \( \cos\theta = p \) and \( \cot\theta = q \), then which of the following relationships is true? (a) \(\cfrac{1}{p^2}+\cfrac{1}{q^2}=1\) (b) \(\cfrac{1}{p^2}-\cfrac{1}{q^2}=1\) (c) \(\cfrac{1}{p^2}-\cfrac{1}{q^2}=0\) (d) \(\cfrac{1}{q^2}+\cfrac{1}{p^2}=1\)

30. If \( \cos^2\theta - \sin^2\theta = \frac{1}{2} \), then what is the value of \( \tan\theta \)? (a) \(-\cfrac{1}{\sqrt3}\) (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(-\cfrac{1}{3}\)

31. If \( r \sin\theta = \frac{7}{2} \) and \( r \cos\theta = \frac{7\sqrt{3}}{2} \), then what are the values of \( r \) and \( \theta \)? (a) \(r=7, \theta=30^0\) (b) \(r=\cfrac{1}{7}, \theta=30^0\) (c) \(r=7, \theta=60^0\) (d) \(r=\cfrac{1}{7}, \theta=60^0\)

32. 1 radian = (a) π/2 right angles (b) π/4 right angles (c) \( \cfrac{4 \, \text{right angles}}{\pi} \) (d) \( \cfrac{2 \, \text{right angles}}{\pi} \)

33. If \( \tanθ + \cotθ = 2 \), the value of \( \tanθ - \cotθ \) will be (a) 2 (b) 0 (c) -2 (d) \(\cfrac{1}{2}\)

34. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \] (a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x-y}{x+y}\) (c) \(\cfrac{x+y}{x-y}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)

35. If \[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, \] then what is the value of \( \theta \)? (a) 60° (b) 30° (c) 45° (d) 90°

36. The minimum and maximum values of \(a + b\sin\theta\) are 5 and 6 respectively. Find the values of \(a\) and \(b\). (a) \(a=1, b=5\) (b) \(a=-5, b=1\) (c) \(a=5, b=1\) (d) \(a=-1, b=5\)

37. In triangle ABC, angle B is a right angle. The hypotenuse is \(\sqrt{15}\), and the sum of the other two sides is 4. What is the value of \((\cos A + \cos C)\)? (a) \(\cfrac{8}{\sqrt{13}}\) (b) \(\cfrac{-8}{\sqrt{15}}\) (c) \(\cfrac{-8}{\sqrt{13}}\) (d) \(\cfrac{8}{\sqrt{15}}\)

38. If \(x = y(\csc\theta + \cot\theta)\) and \(z = y(\csc\theta - \cot\theta)\), then which of the following relationships is correct? (a) \(xy=z^2\) (b) \(xz=y^2\) (c) \(yz=x^2\) (d) \(xyz=1\)

39. If \(\csc\theta + \cot\theta = 2 + \sqrt{3}\), then what is the value of \(\csc\theta - \cot\theta\)? (a) \(\sqrt3+\sqrt2\) (b) \(\sqrt2-3\) (c) \(\sqrt3-2\) (d) \(2-\sqrt3\)

40. "If ???? sin ⁡ ???? = 7 2 and ???? cos ⁡ ???? = 7 3 2 , then what is the value of ???? ? (a) \(49\) (b) \(7\) (c) \(\sqrt7\) (d) \(-7\)

41. If \( 3x = \csc \alpha \) and \( \cfrac{3}{x} = \cot \alpha \), then find the value of \( 3\left(x^2-\cfrac{1}{x^2}\right) \). (a) \(\cfrac{1}{27}\) (b) \(\cfrac{1}{81}\) (c) \(\cfrac{1}{3}\) (d) \(\cfrac{1}{9}\)

42. If 3 cos ⁡ ???? − 4 sin ⁡ ???? = 5 , then what is the value of 3 sin ⁡ ???? + 4 cos ⁡ ???? ? (a) 1 (b) 2 (c) 0 (d) \(\cfrac{1}{2}\)

43. The maximum value of \(5 + 4\sin\theta\) for any value of \(\theta\) will be: (a) 9 (b) 1 (c) 0 (d) 5

44. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)? (a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

45. If \(\cfrac{\sinθ + \cosθ}{\sinθ - \cosθ} = \cfrac{3}{2}\), find the value of \(\cosθ\) (a) \(\cfrac{1}{5}\) (b) \(\cfrac{3}{2}\) (c) \(\cfrac{1}{\sqrt{26}}\) (d) None of these

46. In triangle ABC, \(\angle C = 90^\circ\) and AC : BC = 3 : 4, then what is the value of cosec A? (a) \(\cfrac{3}{4}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{5}{4}\) (d) \(\cfrac{3}{5}\)

47. If \(x\sin45^\circ \cos45^\circ \tan60^\circ = \tan^2 45^\circ - \cos^2 60^\circ\), then what is the value of \(x\)? (a) 1 (b) \(\cfrac{2}{\sqrt3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{\sqrt3}{2}\)

48. If \(0^\circ \leq \alpha < 90^\circ\), find the minimum value of \((\sec^2α + \cos^2α)\). True / False

49. If \( r\cos\theta = 1 \) and \( r\sin\theta = \sqrt{3} \), then the value of \( \theta \) is— (a) \(\cfrac{π}{2}\) (b) \(\cfrac{π}{3}\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{6}\)

50. If \(3x = \csc \alpha\) and \(\cfrac{3}{x} = \cot \alpha\), then find the value of \(3\left(x^2 - \cfrac{1}{x^2}\right)\). (a) \(\cfrac{1}{27}\) (b) \(\cfrac{1}{81}\) (c) \(\cfrac{1}{3}\) (d) \(\cfrac{1}{9}\)

51. If \( \cos\theta - \sin\theta = \sqrt{2} \sin\theta \), then the value of \( \cos\theta + \sin\theta - \sqrt{2} \cos\theta \) is — (a) \(0\) (b) \(1\) (c) \(\sqrt2\) (d) None of the above

52. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then the value of \(x\) is — (a) \(\pm 1\) (b) \(\pm \cfrac{1}{2}\) (c) \(\pm \cfrac{1}{\sqrt2}\) (d) \(\pm \cfrac{1}{\sqrt3}\)

53. If \( \tan\theta + \cot\theta = 2 \), then the value of \( \tan\theta - \cot\theta \) is — True / False

54. If \( \sin\theta \cos\theta = \frac{1}{2} \), then what is the value of \( (\sin\theta + \cos\theta)^2 \)? (a) 1 (b) 3 (c) 2 (d) 4

55. If \(5\cos\theta + 12\sin\theta = 13\), then what is the value of \(\tan\theta\)? (a) \(\cfrac{13}{15}\) (b) \(\cfrac{12}{5}\) (c) \(\cfrac{5}{13}\) (d) \(\cfrac{5}{12}\)

56. If \( \cos^2\theta – \sin^2\theta = \frac{1}{2} \), then what is the value of \( \tan\theta \)? (a) \(\frac{1}{\sqrt3}\) (b) \(\sqrt3\) (c) 1 (d) None of the above

57. If \(tanα + cotα = 2\), then the value of \(tan^{13}α + cot^{13}α\) is—? True / False

58. The minimum value of \( \cos^2\theta + \sec^2\theta \) is: (a) -2 (b) 1 (c) 2 (d) -1

59. If \( \cos\theta + \sin\theta = \sqrt{2} \cos\theta \), then what is the value of \( \sin\theta - \cos\theta \)? (a) \(\sqrt2 sin\theta\) (b) \(-\sqrt2 sin\theta\) (c) \(-\sqrt2 cos\theta\) (d) None of the above

60. If sin ⁡ 3 ???? ⋅ sec ⁡ 6 ???? = 1 , then what is the value of tan ⁡ 6 ???? ? True / False

61. If \(tanθ = 0\), then which of the following is correct? (a) sinθ=0 (b) cosθ=0 (c) cotθ=0 (d) None of these

62. If \(0^\circ < θ < 90^\circ\), then \( \sin θ > \sin^2 θ \). True / False

63. \(\sin 30^\circ + \sin 60^\circ > \sin 90^\circ\) True / False

64. If \( \sqrt{2} \sin(2x + 5^\circ) = \cot 45^\circ \), then what is the value of \( \sec 3x \)?

65. If \(r\cosθ = 1\) and \(r\sinθ = \sqrt{3}\), then find the values of \(r\) and \(θ\).

66. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?

67. If \(\sin^4 \theta + \sin^2 \theta = 1\), then prove that \(\tan^4 \theta - \tan^2 \theta = 1\).

68. If \((x + 1)\cot^2\frac{\pi}{2} = 2\cos^2\frac{\pi}{3} + \frac{3}{4}\sec^2\frac{\pi}{4} + 4\sin^2\frac{\pi}{6}\), then find the value of \(x\).

69. If \(\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 5\), then what is the value of \(\tan \theta\)?

70. Evaluate the expression: \[ \cot^2 30^\circ - 2\cos^2 60^\circ - 4\sin^2 30^\circ - \frac{3}{4}\sec^2 45^\circ + \tan 45^\circ \]

71. If \(\cos^4\theta - \sin^4\theta = \frac{2}{3}\), then what is the value of \(1 - 2\sin^2\theta\)?

72. If \( \cos^2θ − \sin^2θ = \frac{1}{x} \), where \( x > 1 \), then what is the value of \( \cos^4θ − \sin^4θ \)?

73. Given: \(a \cos θ = 3\) and \(b \tan θ = 4\), Find the relation between \(a\) and \(b\) that eliminates \(θ\).

74. Given that \(\cos\alpha = \sin\beta\) and both \(\alpha\) and \(\beta\) are acute angles, find the value of \(\sin(\alpha + \beta)\).

75. If \(x = a\sec\theta\) and \(y = b\tan\theta\), then find the relation between \(x\) and \(y\) eliminating \(\theta\).

76. Left-hand side (LHS): \[ 1 + \cfrac{\tan A}{\tan B} = 1 + \cfrac{\tan(90^\circ - B)}{\tan B} = 1 + \cfrac{\cot B}{\cot B} = 1 + \cot^2 B = \csc^2 B \] Right-hand side (RHS): \[ \tan^2 A \cdot \sec^2 B = \tan^2(90^\circ - B) \cdot \sec^2 B = \cot^2 B \cdot \sec^2 B = \cfrac{\cos^2 B}{\sin^2 B} \cdot \cfrac{1}{\cos^2 B} = \cfrac{1}{\sin^2 B} = \csc^2 B \] \(\therefore\) LHS = RHS (Proved)

77. If \( \sin^2 \theta + 2x \cos^2 \theta = 1 \), then the value of \( x \) will be _____

78. If \[ \frac{\sinθ}{x} = \frac{\cosθ}{y} \] then prove that \[ \sinθ - \cosθ = \frac{x - y}{\sqrt{x^2 + y^2}} \]

79. If \(0^\circ < \theta < 90^\circ\), then \(\sin\theta < \sin^2\theta\).

80. If \(\sin 3x = 1\), what is the value of \(\tan 2x\)?

81. If \(\cos^4\theta - \sin^4\theta = \cfrac{2}{3}\), then find the value of \(1 - 2\sin^2\theta\).

82. From the equation \(5 \sin^2 \theta + 4 \cos^2 \theta = \frac{9}{2}\), find the value of \(\tan \theta\), where \(0^\circ < \theta < 90^\circ\).

83. If \(x = \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

84. Given: \(\tan A = \frac{x}{y}\), find the value of \(\frac{\cos A - \sin A}{\cos A + \sin A}\).

85. If \(x \sin 60^\circ \cos^2 30^\circ = \frac{\tan^2 45^\circ \sec 60^\circ}{\csc 60^\circ}\), find the value of \(x\).

86. If \( \sin x = m \sin y \) and \( \tan x = n \tan y \), then prove that \[ \cos^2 x = \cfrac{m^2 - 1}{n^2 - 1} \]

87. If \( \tan \theta = \cfrac{5}{7} \), then find the value of \[ \cfrac{5\sin \theta + 7\cos \theta}{7\sin \theta + 5\cos \theta}. \]

88. If \( \tan θ + \cot θ = 2 \), then what is the value of \( \tan^7 θ + \cot^7 θ \)?

89. If an arc of a circle measuring 220 cm in length subtends a central angle of 60°, find the radius of the circle.

90. If \( \cos^2 θ - \sin^2 θ = \cfrac{1}{2} \), then find the value of \( \tan^2 θ \).

91. If \(\frac{\sinθ + \cosθ}{\sinθ - \cosθ} = 7\), then what is the value of \(\tanθ\)?

92. If \(\sec \theta = \frac{x}{y}\) and \(x \ne y\), then which one is greater between \(x\) and \(y\)?

93. If \(r \cos \theta = \frac{1}{2}\) and \(r \sin \theta = \frac{\sqrt{3}}{2}\), then find the value of \(r\), where \(0^\circ < \theta < 90^\circ\).

94. AC = CE ABC is an equilateral triangle BC = AC = CE ∠BCA = 60° ∠BCE = 180° − 60° = 120° BC = CE ∠CBE = ∠CEB = \(\frac{180° − 120°}{2} = 30°\) 180° = \(\pi\) radians 120° = \(\frac{\pi × 120}{180} = \frac{2\pi}{3}\) radians 30° = \(\frac{\pi × 30}{180} = \frac{\pi}{6}\) radians Angles of triangle ACE in radians: \(\frac{2\pi}{3}\), \(\frac{\pi}{6}\), \(\frac{\pi}{6}\)

95. If \(\cos\theta = \frac{x}{\sqrt{x^2 + y^2}}\), then prove that \(x \sin \theta = y \cos \theta\).

96. Evaluate the value of: \[ \sin^2 \left(\frac{\pi}{3}\right) - \sec^2 \left(\frac{\pi}{4}\right) - \csc^2 \left(\frac{\pi}{4}\right) + \cot^2 \left(\frac{\pi}{4}\right) \]

97. If \( \cos^2 θ - \sin^2 θ = \frac{1}{2} \), then find the value of \( \cos^4 θ - \sin^4 θ \).

98. If \(\cos \theta = \frac{x}{y}\) and \(x \ne y\), then which one is smaller between \(x\) and \(y\), and why?

99. For which value(s) of \(θ\) (where \(0^\circ ≤ θ ≤ 90^\circ\)) will \(2\sinθ\cosθ = \cosθ\) be true?

100. If \(\sin 4\theta = \cos 5\theta\), find the value of \(\theta\).

101. Prove that \[ \sqrt{\cfrac{1 + \cos \theta}{1 - \cos \theta}} = \csc \theta + \cot \theta \]

102. Find the minimum value of \(9 \tan^2 \theta + 4 \cot^2 \theta\).

103. If \(2 \cos^2\theta + 3 \sin \theta = 3\) and \(0^\circ < \theta < 90^\circ\), find the value of \(\theta\).

104. In triangle △ABC, ∠C = 90°, and if BC = \(m\) and AC = \(n\), then prove that: \[ m \sin A + n \sin B = \sqrt{m^2 + n^2} \]

105. Solve: \[ x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ \]

106. Evaluate the expression: \[ \frac{4}{3} \cot^2 30^\circ + 3 \sin^2 60^\circ - 2 \csc^2 60^\circ - \frac{3}{4} \tan^2 30^\circ \]

107. Determine the values of \(\theta\) for which \(\sin^2\theta - 3\sin\theta + 2 = 0\) holds true, given that \(0^\circ < \theta < 90^\circ\).

108. Eliminate \(\theta\) from the given equations \(2x = 3\sin\theta\) and \(5y = 3\cos\theta\), and express the relationship between \(x\) and \(y\).

109. If \(r \cos \theta = 2\), what should be the value of \(r\) when \(\theta = 60^\circ\)?

110. If \( \tan \theta = \cfrac{8}{15} \), find the value of \( \sin \theta \).

111. The maximum value of sin 3θ is _____ .

112. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

113. Find the value of \(x\) from the equation: \(x \sec^2 45^\circ \cdot \csc^2 45^\circ + 2(\sin 60^\circ + \sin 30^\circ) = \tan 60^\circ\)

114. If \(\cos \theta + \sec \theta = 2\), find the value of \(\cos^9 \theta + \sec^9 \theta\).

115. If \(\sin^2\theta + \sin^4\theta = 1\), then prove that \(\tan^4\theta - \tan^2\theta = 1\).

116. In triangle ∆ABC, ∠B = 90°, AC = √13 cm, and AB + BC = 5 cm. Find the value of (cos A + cos C).

117. If \(\sin^2 x + \sin^2 y = 1\), then what is the value of \(\sin \frac{(x + y)}{2} + \cos \frac{(x + y)}{2}\)?

118. If one angle of a parallelogram is 67°30′, find the radian measures of the other three angles.

119. If \( \csc^2 θ = 2 \cot θ \), then find the value of \( θ \), where \( 0^\circ < θ < 90^\circ \).

120. If \(\sin 17^\circ = \frac{x}{y}\), then show that \(\sec 17^\circ - \sin 73^\circ = \frac{x^2}{y\sqrt{y^2 - x^2}}\)

121. Evaluate the value of: \[ \cfrac{5\cos^2\left(\cfrac{\pi}{3}\right) + 4\sec^2\left(\cfrac{\pi}{6}\right) - \tan^2\left(\cfrac{\pi}{4}\right)}{\sin^2\left(\cfrac{\pi}{6}\right) + \cos^2\left(\cfrac{\pi}{6}\right)} \]

122. If \(\sin \theta = \cfrac{p^2 - q^2}{p^2 + q^2}\), then show that \(\cot \theta = \cfrac{2pq}{p^2 - q^2}\) where \(p > q\) and \(0^\circ < \theta < 90^\circ\).

123. Find the value of: \( \sec^2 45^\circ - \cot^2 45^\circ - \sin^2 30^\circ - \sin^2 60^\circ \)

124. Evaluate the expression: \(\cot^2 30^\circ - 2\cos^2 60^\circ - \cfrac{3}{4}\sec^2 45^\circ - \sin^2 30^\circ\)

125. Determine the value of: \[ \cfrac{4}{1+\tan^2\theta}+\cfrac{3}{1+\cot^2\theta}+\sin^2\theta \]

126. How many degrees is \(\frac{\pi}{12}\) radians equal to?

127. If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]

128. If \( \cosθ = \frac{x}{\sqrt{x^2 + y^2}} \), then prove that \( x\sinθ = y\cosθ \).

129. What is the value of \[ \frac{4}{\sec^2θ} + \frac{1}{1 + \cot^2θ} + 3\sin^2θ? \]

130. If \( \csc \theta + \cot \theta = \sqrt{3} \), then find the value of \( \sin \theta \), where \( 0^\circ < \theta < 90^\circ \).

131. Evaluate the expression: \[ \frac{1 - \sin^2 30^\circ}{1 + \sin^2 45^\circ} \times \frac{\cos^2 60^\circ + \cos^2 30^\circ}{\csc^2 90^\circ - \cot^2 90^\circ} \div (\sin 60^\circ \tan 30^\circ) \]

132. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).

133. If \(∠A + ∠B = 90°\), then find the value of \(1 + \frac{\tan A}{\tan B}\).

134. Find the value of: \[ \frac{2\tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ - \sec 60^\circ \]

135. If \[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3 \] then find the value of \[ \sin^4 \theta - \cos^4 \theta \]

136. In triangle XYZ, ∠Y is a right angle. Given: XY = \(2\sqrt{6}\) and XZ − YZ = 2 Find the value of sec X + tan X.

137. If \(\sec^2 \theta + \tan^2 \theta = \frac{13}{12}\), then what is the value of \(\sec^4 \theta - \tan^4 \theta\)?

138. Given: \[ x \cos \theta = 3 \quad \text{and} \quad 4 \tan \theta = y \] Find the relation between \(x\) and \(y\) eliminating \(\theta\).

139. Find the value of \( \sin^6 \alpha + \cos^6 \alpha + 3\sin^2 \alpha \cos^2 \alpha \).

140. Given: \(\tan \theta + \sin \theta = m\) and \(\tan \theta - \sin \theta = n\) Prove that: \[ m^2 - n^2 = 4\sqrt{mn} \quad \text{where } 0^\circ < \theta < 90^\circ \]

141. Evaluate the expression: \[ \frac{(\sin 0^\circ + \sin 60^\circ)(\cos 60^\circ + \cot 45^\circ)}{(\cot 60^\circ + \tan 30^\circ)(\csc 30^\circ - \csc 90^\circ)} \]

142. Given: \(\cos \alpha = \frac{5}{13}\) and \(0^\circ < \alpha < 90^\circ\) Prove that: \(\cot \alpha + \csc \alpha = 1.5\)