CLASS 12 INVERSE TRIGNOMETERY DPP
CH2 - INVERSE TRIGNOMETRIC
✤ Very Short (Objective Type) / Short Answer Type
Q1. If sin⁻¹x = π/10, for some x∈R, then the value of cos⁻¹x is _____.
Q2. If tan⁻¹x - cot⁻¹x = π/6, then x is _____.
Q3. The domain of the function y = sin⁻¹(-x²) is
(a) [0, 1] (b) (0, 1)
(c) [-1, 1] (d) φ
Q4. If sec⁻¹x + sec⁻¹y = π/2, the value of cosec⁻¹x + cosec⁻¹y is
(a) π (b) π/2
(c) 3π/2 (d) ≥ -π
Q5. The value of tan²(sec⁻¹2) + cot²(cosec⁻¹3) is
(a) 5 (b) 11
(c) 13 (d) 15
Q6. cot(π/4 - 2 cot⁻¹3) = 7, state true or false.
Q7. If 3 sin⁻¹x + cos⁻¹x = π, then x is equal to
(a) 0 (b) 1/√2
(c) -1 (d) 1/2
Q8. Show that
sin⁻¹(2x√(1-x²)) = 2 sin⁻¹x, - 1/√2 ≤ x ≤ 1/√2
[NCERT]
Q9. Find the value of sec (tan-12). [NCERT Exemplar]
Q10. If sin(sin-11/5+cos-1x) = 1, then find the value of x.
[NCERT; HOTS; Delhi 2014]
Q11. Find the value of tan-1 (x/y) - tan-1 ((x-y)/(x+y)).
[NCERT; Delhi 2013 (C), 2011]
Q12. If y = cot-1 (√cos x) - tan-1(√cos x), then prove that
sin y = tan² (x/2).
[Foreign 2013]
Q13. Prove that 3 sin-1x = sin-1(3x-4x3), x ∈ [-1/2, 1/2].
[CBSE 2018]
Q14. Prove that sin-1x = tan-1 (x/√(1-x²)).
Q15. Write cot-1 (1/√(x²-1)), |x| > 1 in the simplest form.
[Foreign 2013]
Q16. If sin-1x + sin-1y = 2π/3, then find the value of cos-1x + cos-1y.
Q17. Find the value of tan(2 tan⁻¹(1/5)). [Delhi 2013]
✤ Long Answer I / Long Answer II Type
Q18. Prove that
tan(π/4 + 1/2 cos⁻¹(a/b)) + tan(π/4 - 1/2 cos⁻¹(a/b)) = 2b/a
[DoE; Delhi 2017]
Q19. Prove that sin⁻¹(⁴/₅) - sin⁻¹(⁵/₁₃) - sin⁻¹(¹⁶/₆₅) = π/2
Q20. Prove the following: tan⁻¹√x = ½ cos⁻¹((1-x)/(1+x)) x ∈ [0, 1]
[NCERT]
Q21. Prove that tan⁻¹((√1+x-√1-x)/(√1+x+√1-x)) =
π/4 - ½ cos⁻¹x, -1/√2 ≤ x ≤ 1
[NCERT; DoE: AI 2011]
Q22. Prove the following:
cos[tan⁻¹(sin(cot⁻¹x))] = √(1+x²)/√(2+x²)
Q23. Prove the following:
cos(sin⁻¹(3/5)+cot⁻¹(3/2)) = 6/(5√13)
Q24. If x = a cos 0 + b sin 0, y = a sin 0 - b cos 0, prove that
x²/a² + y²/b² = 1.
[MOTO, 2008]
Q25. Show that tan (½ sin⁻¹ 3/5) = 4 - √7/3. [IIT 2003]
Q26. Prove That sec² (tan⁻¹ 2) - cosec² (cot⁻¹ 3) = 5.
Q27. If a₁, a₂, a₃, ..., aₙ is an arithmetic progression with common
difference d, then evaluate the following:
tan [ tan⁻¹ ( d / (1+ a₁ a₂)) - tan⁻¹ ( d / (1+ a₂ a₃)) –
tan⁻¹ ( d / (1+ a₃ a₄)) - tan⁻¹ ( d / (1+ aₙ-₁ aₙ) ) ]
Q28. Prove that tan⁻¹ (1/5) - sec⁻¹ (√(5√2)/7) - tan⁻¹(1/8) = π / 4
Q29. Write the following function in the simplest form:
tan⁻¹ (cos x / (1 + sin x))
Q30. Simplify:
tan⁻¹ ((a cos x - b sin x) / (b cos x + a sin x)), if a/b tan x > -1.
[NCERT]
Q31. Write the following function in the simplest form:
Q32. Solve for x, tan⁻¹ ((1-x)/(1+x)) = 1/2 tan⁻¹(x), x > 0.
[NCERT; NCERT Exemplar; Foreign 2011]
Q33. Solve the following equation:
tan⁻¹ ((x+1)/(x-1)) + tan⁻¹ ((x-1)/(x-1)) = tan⁻¹ (-7).
Q34. Write the following function in the simplest form:
tan-1(cos x - sin x)/(cos x + sin x), 0<x<π [NCERT]
Q35. If tan-1(x-1)/(x-2) + tan-1(x+1)/(x+2) = π/4, then find the
value of x. [NCERT; Foreign 2013]
Q36. Solve the following equation:
cos (tan-1x) = sin (cot-13/4)
[NCERT Exemplar; Delhi 2017; Foreign 2014; A1 2013]
Q37. Find the value of the expression
sin (2 tan⁻¹ 1/5) - cos (tan⁻¹ 2√2).
Q38. Find the value of cot 1/2 [cos⁻¹ 2x/1+x² + sin⁻¹ 1-y²/1+y²]
|x| < 1, y > 0 and xy < 1.
[Foreign 2017]
Q39. Prove that: tan⁻¹( (√1+x² + √1-x²) / (√1+x² - √1-x²) )
Q40. cos⁻¹(cos 13π/6).
Q41. Write the principal value of the following: sin⁻¹(sin 4π/5).
Q42. Using principal value evaluate the following:
cos⁻¹(cos 2π/3) + sin⁻¹(sin 2π/3). [AI 2011]
Q43. If sin⁻¹x + sin⁻¹y = π, then find the values of x and y.
Q44. If x < y < 0, such that xy = 1, then find the value of tan⁻¹x + tan⁻¹y.
Prove that in Question 45 to 55:
Q45. sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16) = π. [NCERT]
Q46. cot⁻¹√((1 + sin x) + √(1 - sin x))/((1 + sin x) - √(1 - sin x)) = x/2,
x ∈ (0, π/4) [NCERT; Delhi 2014, 11]
Q47. tan⁻¹(1/4) + tan⁻¹(2/9) = 1/2 cos⁻¹(3/5) [HOTS]
Q48. cos⁻¹(12/13) + sin⁻¹(3/5) = sin⁻¹(56/65)
[NCERT; Dehradun 2019]
Q49. tan⁻¹2x + tan⁻¹(4x/(1-4x²)) = tan⁻¹((6x-8x³)/(1-12x²)),
|x| < 1/(2√3) [Foreign 2017]
Q50. tan⁻¹(1/2) + tan⁻¹(1/7) = tan⁻¹(31/17) [NCERT; AI 2011]
Q51. sin⁻¹(8/17) + sin⁻¹(3/5) = cos⁻¹(36/85) = tan⁻¹(77/36)
[NCERT; Delhi 2012, 13 (C)]
Q52. sin⁻¹(1/√5) + cot⁻¹(3) = π/4. [HOTS]
Q53. cos(sin⁻¹(3/5) + sin⁻¹(5/13)) = 33/65.
Q54. cos⁻¹(4/5) + cos⁻¹(12/13) = cos⁻¹(33/65). [NCERT]
Q55. 2 sin⁻¹(3/5) = tan⁻¹(24/7).
Write the following functions in the simplest form (Question 56 and 27):
Q56. tan⁻¹(3x - x³)/(1 - 3x²)
Q57. tan⁻¹√((1 - cos 3x)/(1 + cos 3x)), x < π [NCERT]
Q58. Solve for x, tan⁻¹(x + 1) + tan⁻¹(x - 1) = tan⁻¹(8/31)
[Foreign 2015]
Q59. Find the value of x, if sin [cot⁻¹(x + 1)] = cos (tan⁻¹x)
[DoE; Bhubaneshwar 2015, Delhi 2015]
Q60. Prove the following: 2 sin⁻¹(3/5) - tan⁻¹(17/31) = π/4
[Bhubaneshwar 2015]
Q61. Solve the following for x:
tan⁻¹((x - 2)/(x - 3)) + tan⁻¹((x + 2)/(x + 3)) = π/4, |x| < 1
[Patna 2015]
Q62. Prove the following: sin [tan⁻¹((1 - x²)/2x)] + cos⁻¹((1 - x²)/
(1 + x²)) = 1, 0 < x < 1 [Guwahati 2015]
Q63. Solve for x, 2 tan⁻¹(sin x) = tan⁻¹(2 sec x), x ≠ π/2
[DoE; Foreign 2012]
Q64. Solve for x: tan⁻¹(2x/(1-x²)) + cot⁻¹((1-x²)/2x) = 2π/3, x > 0.
Q65. Solve for x: cos⁻¹((x² - 1)/(x² + 1)) + 1/2 tan⁻¹(2x/(1-x²)) = 2π/3
Q66. Solve for x: sin⁻¹(2α/(1 + α²)) + sin⁻¹(2β/(1 + β²)) = 2 tan⁻¹x.
[HOTS]
Q67. Solve for x: sin⁻¹x + sin⁻¹2x = π/3 [HOTS]
✤ INTEGRATED EXERCISE
Very Short (Objective Type) / Short Answer Type
1. The principal value of sin⁻¹(sin 2π/3) is
(a) 2π/3 (b) π/3 (c) -π/6 (d) π/6
2. The value of cos⁻¹(1/2) + 3 sin⁻¹(1/2) is equal to
(a) π/4 (b) π/6 (c) 2π/3 (d) 5π/6
3. The greatest and least values of (sin⁻¹x)² + (cos⁻¹x)² are respectively
____________.
4. The value of sin(2 sin⁻¹ (-6)) is ______.
5. Find the principal value of cosec⁻¹(2). [NCERT]
6. Evaluate tan⁻¹(sin (-π/2)). [NCERT Exemplar]
7. Write the value of cos⁻¹(-1/2) + 2 sin⁻¹(1/2).
8. Write one branch of sin⁻¹x other than the principal branch.
9. Find the principal value of tan⁻¹ (-1)
10. Find the principal value of cos⁻¹(cos(7π/6))
11. Find the value of sin(2 sin⁻¹(3/5)).
12. Find the principal value of tan⁻¹(tan(9π/8))
13. Write the principal value of tan⁻¹(tan(3π/4)).
✤ Long Answer I / Long Answer II Type
14. Prove that tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) = π
15. Prove the following: (9π/8) - (9/4) sin⁻¹(1/3) = (9/4) - sin⁻¹(2√2/3)
16. Show that
2 tan⁻¹ {tan(α/2) tan(π/4 - β/2)} = tan⁻¹((sin α cos β)/(cos α + sin β)
17. Write the following function in the simplest form:
sin⁻¹(x√1-x - √x√1-x²)
18. Solve the following for x:
cos⁻¹((x²-1)/(x²+1)) + tan⁻¹((2x)/(x²-1)) = 2π/3
19. Prove that sin⁻¹(63/65) = sin⁻¹(5/13) + cos⁻¹(3/5).
20. Find the value of the following:
tan(1/2)sin⁻¹((2x)/(1+x²)) + cos⁻¹((1-y²)/(1+y²)), |x| < 1, y > 0
and xy < 1.
21. If cos⁻¹(x/2) + cos⁻¹(y/3) = 0, then prove that
9x² - 12xy cos 0 + 4y² = 36 sin²0.
22. Evaluate cot(√(1+x²) -x)
23. Solve the following equation: sin⁻¹(1-x) - 2 sin⁻¹x = π/2
24. Evaluate tan [2 tan⁻¹(1/5) + π/4]
25. Solve for x, tan⁻¹(2x) + tan⁻¹(3x) = π/4
26. Solve for x: tan⁻¹√(1+x) -√(1-x) / √(1+x) + √(1-x) = β.
27. Find the solution of the equation tan⁻¹x - cot⁻¹x = tan⁻¹(1/√3).
28. If tan⁻¹(1+1.2) + tan⁻¹(1/1+2.3) + ... + tan⁻¹(1/1+n(n+1)) = tan⁻¹0,
then find the value of θ.
29. Find the principal value of tan⁻¹(tan(5π/6))
30. Prove that tan⁻¹(cos x / 1-sin x) - cot⁻¹(1+cos x / 1-cos x) = π/4,
x ∈ (0, π/2).
31. Evaluate tan [1/2 cos⁻¹(√3/11)]
32. If tan⁻¹a + tan⁻¹b + tan⁻¹c = π, then prove that a + b + c = abc.
ASSESS YOURSELF
1. The equation tan⁻¹x - cot⁻¹x = tan⁻¹(1/√3) has solution as _______.
2. If α ≤ 2 sin⁻¹x + cos⁻¹x ≤ β, then α = _______.
3. If tan⁻¹x = π/10 for some x ∈ R, then the value of cot⁻¹x is
(a) π/5, (b) 2π/5, (c) 3π/5, (d) 4π/5
4. Show that sin⁻¹((a-x)/2a) = 1/2 cos⁻¹x/a.
Write the principal values in Exercises 5 to 8:
5. cosec⁻¹(2) 6. cos⁻¹(-√3/2)
7. tan⁻¹(-√3) 8. tan⁻¹(tan(3π/4))
Write the value in Exercises 9 to 11:
9. cosec⁻¹(√2) + sec⁻¹(√2)
10. cos⁻¹(cos(2π/3)) + sin⁻¹(cos(2π/3))
11. tan⁻¹(√3) + cot⁻¹(1/√3)
12. What is the domain of the function cosec⁻¹ x?
13. Write one branch of tan⁻¹ x other than the principal branch.
Evaluate in Exercises 14 to 20:
14. sin⁻¹ {cos (sin⁻¹(1/2))}
15. cosec⁻¹ {cosec (-π/4)}
16. cos {π/3 - cos⁻¹(1/2)}
17. sec²(tan⁻¹2) 18. cos⁻¹(cos 5π/3)
19. sec⁻¹ ((x-3)/(x+3)) + sin⁻¹ ((x+3)/(x-3))
20. tan⁻¹{cos π}
Prove that in Exercises 21 to 24:
21. 2 tan⁻¹x = sin⁻¹(2x)/(1+x²)
22. 2 cos⁻¹x = sec⁻¹(1)/(2x²-1)
23. sin⁻¹x = cot⁻¹(√(1-x²))/x)
24. cos⁻¹x = 2 cos⁻¹√(1+x)/2
25. Find the value of cosec(cot⁻¹y/2) in terms of y alone.
Prove that (Exercises 26 to 34):
26. 2 tan⁻¹(1/5) + tan⁻¹(1/8) = tan⁻¹(4/7)
27. tan⁻¹(1/5) + tan⁻¹(1/3) + tan⁻¹(1/7) + tan⁻¹(1/8) = π/4
28. 2 tan⁻¹(1/5) + tan⁻¹(1/8) = tan⁻¹(4/7)
29. cos⁻¹(4/5) + cos⁻¹(12/13) = cos⁻¹(33/65)
30. cos⁻¹(4/5) + tan⁻¹(3/5) = tan⁻¹(27/11)
31. sin⁻¹(1/4) + 2 tan⁻¹(1/3) = π/2
32. tan⁻¹(3a²x-x³)/(a(a²-3x²)) = 3 tan⁻¹(x/a)
33. sec²(tan⁻¹3) + cosec²(cot⁻¹4) = 27
34. sin⁻¹((x + √(1-x²))/√2) = π/4 + sin⁻¹x, -1 ≤ x ≤ 1.
Write in the simplest form (Exercises 35 to 38):
35. cos⁻¹(3/5 cos x + 4/5 sin x)
36. tan⁻¹(8x)/(1+20x²)
37. cot⁻¹(√(1 + cos 5x)/(1 - cos 5x))
38. sin⁻¹(x√(1-x²) + x√(1-x⁴))
Solve for x (Exercises 39 to 45):
39. cos⁻¹x + sin⁻¹x = π/2
40. tan(cos⁻¹x) = sin(cot⁻¹(1/2))
41. cot⁻¹(2/x) + cot⁻¹(3/x) = π/4
42. tan⁻¹((x-3)/(x-4)) + tan⁻¹((x+3)/(x+4)) = π/4
[AI 2017]
43. 2 cot⁻¹x + tan⁻¹x = 2π/3
44. sin⁻¹(3x/5) + sin⁻¹(4x/5) = sin⁻¹x
45. tan⁻¹√x² + x + sin⁻¹√x² + x + 1 = sin⁻¹1
46. If cos⁻¹a + cos⁻¹b + cos⁻¹c = π, prove that
a² + b² + c² + 2abc = 1.
47. If tan⁻¹a + tan⁻¹b + tan⁻¹c = π/2, prove that ab + bc + ca = 1.
48. Show that
2 tan⁻¹(tan α/2 tan β/2) = cos⁻¹((cos α + cos β)/(1 + cos α cos β))
49. If sin⁻¹x + sin⁻¹y + sin⁻¹z = π, then prove that
x√(1-x²) + y√(1-y²) + z√(1-z²) = 2xyz.
50. If sin⁻¹(2a)/(1+a²) - cos⁻¹(1-b²)/(1+b²) = tan⁻¹(2x)/(1-x²), prove
That x = (a-b)/(1+ab)
51. Find the value of sin{2 cot⁻¹(-5/12)}.
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