Inverse Trigonometric Functions

Inverse Trigonometric Functions

📘 Class 11 & 12 Mathematics Inverse Trigonometric Functions

Chapter: Inverse Trigonometric Functions

Complete Grand master Bikram Sutradhar-Level Question Bank (Only Questions)

🔹 ONE MARK QUESTIONS (1–20)

1. Find the value of (i) $ \sin^{-1}\frac{1}{2} $ (ii) $ \cos^{-1}0 $ (iii) $ \tan^{-1}1 $ (iv) $ \cot^{-1}1 $
2. Evaluate: (i) $ \sin\left(2\sin^{-1}\frac{3}{5}\right) $ (ii) $ \tan\left(\cos^{-1}\frac{4}{5}\right) $
3. If $ \tan^{-1}x + \tan^{-1}y = \frac{\pi}{4} $ and $ xy < 1 $, find $ x+y+xy $.
4. Solve $ 3\tan^{-1}x + \cot^{-1}x = \pi $ for $ x $.
5. If $ \sin^{-1}x + \sin^{-1}y = \frac{\pi}{3} $, find $ \cos^{-1}x + \cos^{-1}y $.
6. If $ \cos^{-1}a + \cos^{-1}b + \cos^{-1}c = \pi $, find $ a+b+c $.
7. Evaluate $ \tan^{-1}(\sec 2) + \cot^{-1}(\csc 3) $.
8. Evaluate $ \sin^{-1}\Big[\cot^{-1}(\cos^{-1}(\tan 1))\Big] $.
9. If $ a = 2\sin^{-1}x + \cos^{-1}x $, find constants $ a $ .
10. Solve $ \cos^{-1}(\sin(\cos x)) = \frac{\pi}{4} $.
11. Find the principal value of $ \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) $.
12. Evaluate $ \cos\left(2\cos^{-1}\frac{2}{3}\right) $.
13. Solve for $ x $: $ \tan^{-1}x – \tan^{-1}2x = 0 $.
14. If $ \sin^{-1}x – \sin^{-1}y = \frac{\pi}{6} $, find $ x-y $ in terms of $ x $ and $ y $.
15. Find the value of $ \tan\Big(\sin^{-1}\frac{4}{5}\Big) $.
16. Evaluate $ \cos^{-1}(\sin(\frac{\pi}{4})) $.
17. If $ \cot^{-1}x + \cot^{-1}y = \frac{\pi}{2} $, find $ xy $.
18. Solve $ \tan^{-1}2 + \tan^{-1}3 $ in exact form.
19. Find $ x $ if $ 2\tan^{-1}x = \frac{\pi}{3} $.
20. Evaluate $ \sin^{-1}(\cos \frac{\pi}{3}) $.

🔹 TWO MARK QUESTIONS (21–40)

21. Prove $ \tan^{-1}\frac{1}{2}\sin^{-1}\frac{3}{4} = \frac{1}{2}\tan^{-1}\frac{12}{5} $.
22. Show that $ \sin^{-1}(\cot^{-1}(\cos(\tan x))) $ is independent of $ x $.
23. Evaluate $ \tan^{-1}\frac{2}{3} + \tan^{-1}\frac{1}{5} $.
24. Solve $ \tan^{-1}x + \tan^{-1}2x = \frac{\pi}{4} $.
25. Solve $ \tan^{-1}(2x) – \tan^{-1}(x) = \frac{\pi}{6} $.
26. Prove $ \sin^{-1} \frac{1}{3} + \sin^{-1} \frac{1}{2} = \sin^{-1} \frac{7}{12} $.
27. Solve for $ x $: $ \tan^{-1}x + \tan^{-1}(x+1) = \frac{\pi}{4} $.
28. If $ \sin^{-1}x + \sin^{-1}y = \frac{\pi}{2} $, find $ x^2 + y^2 $.
29. Evaluate $ \cos^{-1}\frac{1}{\sqrt{2}} + \sin^{-1}\frac{1}{\sqrt{2}} $.
30. Solve $ \tan^{-1}x + \tan^{-1}2x + \tan^{-1}3x = \pi $.
31. If $ \sin^{-1}x = \cos^{-1}y $, find relation between $ x $ and $ y $.
32. Prove $ \tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 = \pi $.
33. Find the value of $ \sin(2\tan^{-1}3) $.
34. Evaluate $ \tan^{-1}5 – \tan^{-1}2 $ in exact form.
35. Solve $ 2\tan^{-1}x = \tan^{-1}\frac{3}{4} $.
36. If $ \sin^{-1}x + \sin^{-1}y = \frac{\pi}{6} $, express $ y $ in terms of $ x $.
37. Evaluate $ \cos^{-1}(\sin(\frac{\pi}{4})) $.
38. Solve $ \tan^{-1}x + \tan^{-1}\frac{1}{x} = \frac{\pi}{2} $.
39. Find $ x $ if $ \cot^{-1}x + \tan^{-1}x = \frac{\pi}{4} $.
40. Evaluate $ \sin^{-1}\frac{3}{5} + \cos^{-1}\frac{4}{5} $.

🔹 FOUR MARK QUESTIONS (41–70)

41. Prove $ \sin^{-1}\frac{1}{4} + \sin^{-1}\frac{1}{5} = \sin^{-1}\frac{9}{20} $.
42. Prove $ \cot^{-1}7 + \cot^{-1}8 + \cot^{-1}18 = \cot^{-1}3 $.
43. Prove $ 2\tan^{-1}\frac{\sin x}{1+\cos x} = x $.
44. Solve $ \tan^{-1}x + \tan^{-1}(x+2) = \frac{\pi}{4} $.
45. Prove $ \tan^{-1}a + \tan^{-1}b = \tan^{-1}\frac{a+b}{1-ab}, \quad ab<1 $.
46. Solve $ \tan^{-1}x + \tan^{-1}(x-1) = \frac{\pi}{6} $.
47. Prove $ \sin^{-1}x + \sin^{-1}y = \sin^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2}) $.
48. Solve $ 2\tan^{-1}x + \tan^{-1}(2x) = \frac{\pi}{2} $.
49. Evaluate $ \tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 + \tan^{-1}4 $.
50. Prove $ \cos^{-1}x + \cos^{-1}y = \cos^{-1}(xy – \sqrt{(1-x^2)(1-y^2)}) $.
51. Solve $ \tan^{-1}x – \tan^{-1}(x-1) = \frac{\pi}{4} $.
52. Find the value of $ \sin(2\sin^{-1}0.6) $ using identities.
53. Solve $ \tan^{-1}x + \tan^{-1}2x + \tan^{-1}3x = \pi $.
54. Prove $ \tan^{-1}3 + \tan^{-1}7 = \tan^{-1}5 $.
55. Solve $ \sin^{-1}x + \sin^{-1}y = \frac{\pi}{3} $.
56. Prove $ \tan^{-1}1 + \tan^{-1}\frac{1}{2} = \frac{\pi}{4} $.
57. Solve $ \tan^{-1}(x+1) – \tan^{-1}x = \frac{\pi}{6} $.
58. Evaluate $ \cos(2\tan^{-1}\frac{1}{2}) $.
59. Solve $ \tan^{-1}x + \tan^{-1}(2x+1) = \frac{\pi}{4} $.
60. Prove $ \tan^{-1}x + \tan^{-1}\frac{1}{x} = \frac{\pi}{2}, \quad x>0 $.
61. Solve $ \sin^{-1}x – \sin^{-1}y = \frac{\pi}{4} $.
62. Evaluate $ \tan(2\tan^{-1}3) $.
63. Solve $ \tan^{-1}x + \tan^{-1}(x+3) = \frac{\pi}{3} $.
64. Prove $ 2\tan^{-1}x + \cot^{-1}x = \pi $.
65. Solve $ \sin^{-1}x + \cos^{-1}y = \frac{3\pi}{4} $.
66. Prove $ \tan^{-1}x – \tan^{-1}y = \tan^{-1}\frac{x-y}{1+xy} $.
67. Solve $ \tan^{-1}x + \tan^{-1}2x + \tan^{-1}3x + \tan^{-1}4x = \pi $.
68. Find domain and range of $ f(x) = \tan^{-1}(\frac{2x}{1-x^2}) $.
69. Solve $ \tan^{-1}x + \tan^{-1}\frac{1-x}{1+x} = \frac{\pi}{4} $.
70. Evaluate $ \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{5} $.

🔹 HIGHER ORDER THINKING (HOTS) QUESTIONS (71–90)

71. Solve $ \tan^{-1}\frac{1}{x} + \tan^{-1}\frac{1}{x+1} = \frac{\pi}{4} $.
72. Prove $ \sin^{-1}x + \sin^{-1}y = \sin^{-1}(x\sqrt{1-y^2} + y\sqrt{1-x^2}) $.
73. Find domain and range of $ f(x) = \tan^{-1}\frac{2x}{1-x^2} $.
74. Prove $ \cos^{-1}x + \cos^{-1}y = \cos^{-1}(xy – \sqrt{(1-x^2)(1-y^2)}) $.
75. Solve $ 2\tan^{-1}x + \tan^{-1}(1-x) = \frac{\pi}{2} $.
76. If $ \sin^{-1}x + \sin^{-1}y = \frac{\pi}{2} $, find $ xy $ in terms of $ x $ and $ y $.
77. Evaluate $ \tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 + \tan^{-1}4 + \tan^{-1}5 $.
78. Prove $ \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \tan^{-1}\frac{5}{7} $.
79. Solve $ \tan^{-1}x + \tan^{-1}2x + \tan^{-1}3x = \pi $ for real $ x $.
80. Evaluate $ \sin^{-1}\frac{1}{3} + \sin^{-1}\frac{2}{3} $.
81. Solve $ \tan^{-1}x + \tan^{-1}(x+4) = \frac{\pi}{4} $.
82. Prove $ 2\tan^{-1}\frac{\sin x}{1+\cos x} = x $ for all $ x \in \mathbb{R} $.
83. Solve $ \tan^{-1}x – \tan^{-1}y = \tan^{-1}\frac{x-y}{1+xy} $.
84. Find domain and range of $ f(x) = \sin^{-1}(\frac{2x}{1+x^2}) $.
85. Solve $ \tan^{-1}x + \tan^{-1}(2x-1) = \frac{\pi}{3} $.
86. Evaluate $ \sin(2\tan^{-1}\frac{1}{3}) $.
87. Solve $ \tan^{-1}x + \tan^{-1}\frac{1}{x} = \frac{\pi}{2}, \quad x>0 $.
88. Solve $ \tan^{-1}x + \tan^{-1}(x+1) + \tan^{-1}(x+2) = \pi $.
89. Prove $ \tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 = \pi $.
90. Solve $ \tan^{-1}x + \tan^{-1}(x+5) = \frac{\pi}{4} $.

🔹 COMPETITIVE TYPE QUESTIONS (91–110)

91. If $ \sin^{-1}x + \sin^{-1}y = \frac{\pi}{2} $, prove $ x^2 + y^2 + 2xy\sqrt{(1-x^2)(1-y^2)} = 1 $.
92. Solve $ \tan^{-1}x + \tan^{-1}2x + \tan^{-1}3x + \tan^{-1}4x + \tan^{-1}5x = \pi $.
93. Prove $ \sin^{-1}\frac{1}{4} + \sin^{-1}\frac{1}{3} = \sin^{-1}\frac{7}{12} $.
94. Solve $ \tan^{-1}x + \tan^{-1}2x + \tan^{-1}3x = \frac{\pi}{2} $.
95. Evaluate $ \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{5} + \tan^{-1}\frac{1}{8} $.
96. Prove $ \tan^{-1}a + \tan^{-1}b + \tan^{-1}c = \pi $ given $ a+b+c=3 $.
97. Solve $ \tan^{-1}x + \tan^{-1}(x+6) = \frac{\pi}{4} $.
98. Prove $ \tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} = \tan^{-1}\frac{1}{2} $.
99. Solve $ \tan^{-1}x + \tan^{-1}(x+7) + \tan^{-1}(x+14) = \pi $.
100. If $ \sin^{-1}x + \cos^{-1}y = \frac{\pi}{2} $, express $ y $ in terms of $ x $.
101. Prove $ \tan^{-1}x + \tan^{-1}\frac{1}{x} = \frac{\pi}{2}, \quad x>0 $.
102. Prove $ 2\tan^{-1}\frac{1}{1+x} = \tan^{-1}\frac{2}{x} $.
103. Prove $ \sin^{-1}x + \sin^{-1}y = \sin^{-1}(x\sqrt{1-y^2} + y\sqrt{1-x^2}) $.
104. Prove $ \cos^{-1}x + \cos^{-1}y = \cos^{-1}(xy – \sqrt{(1-x^2)(1-y^2)}) $.
105. Prove $ \tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 = \pi $.
106. Prove $ \tan^{-1}a + \tan^{-1}b = \tan^{-1}\frac{a+b}{1-ab}, \quad ab<1 $.
107. Solve $ \tan^{-1}x + \tan^{-1}2x = \frac{\pi}{4} $.
108. Solve $ \tan^{-1}x – \tan^{-1}(x-1) = \frac{\pi}{4} $.
109. Prove $ 2\tan^{-1}\frac{\sin x}{1+\cos x} = x $.
110. Prove $ \sin^{-1}\frac{1}{4} + \sin^{-1}\frac{1}{5} = \sin^{-1}\frac{9}{20} $.

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Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar

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