Inverse Trigonometric Identities

Last Updated : 12 Sep, 2026

Before proceeding, make sure you are familiar with inverse trigonometric functions.

Inverse trigonometric identities involve inverse trigonometric functions such as sin⁡-1(x), cos-1(x), and tan⁡-1(x), etc. These functions provide the angles (or arcs) corresponding to a given trigonometric ratio.

Some important identities are:

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The inverse trigonometric identities help in simplifying complex expressions and solving equations involving trigonometric functions.

Properties of Inverse Trigonometric Functions

The following are the properties of inverse trigonometric functions:

Property 1:

  • sin-1 (1/x) = cosec-1 x, for x ≥ 1 or x ≤ -1
  • cos-1 (1/x) = sec-1 x, for x ≥ 1 or x ≤ -1
  • tan-1 (1/x) = cot-1 x, for x > 0

Property 2:

  • sin-1 (-x) = -sin-1 x, for x ∈ [-1 , 1]
  • tan-1 (-x) = -tan-1 x, for x ∈ R
  • cosec-1 (-x) = -cosec-1 x, for |x| ≥ 1

Property 3

  • cos-1 (-x) = π - cos-1 x, for x ∈ [-1 , 1]
  • sec-1 (-x) = π - sec-1 x, for |x| ≥ 1
  • cot-1 (-x) = π - cot-1 x, for x ∈ R

Property 4

  • sin-1 x + cos-1 x = π/2, for x ∈ [-1,1]
  • tan-1 x + cot-1 x = π/2, for x ∈ R
  • cosec-1 x + sec-1 x = π/2 , for |x| ≥ 1

Property 5

  • tan-1 x + tan-1 y = tan-1 ( x + y )/(1 - xy), for xy < 1
  • tan-1 x - tan-1 y = tan-1 (x - y)/(1 + xy), for xy > -1
  • tan-1 x + tan-1 y = π + tan-1 (x + y)/(1 - xy), for xy >1 ; x, y >0

Property 6

  • 2tan-1 x = sin-1 (2x)/(1 + x2), for |x| ≤ 1
  • 2tan-1 x = cos-1 (1 - x2)/(1 + x2), for x ≥ 0
  • 2tan-1 x = tan-1 (2x)/(1 - x2), for -1 < x <1

Identities of Inverse Trigonometric Function

The following are the identities of inverse trigonometric functions:

  1. sin-1 (sin x) = x provided -π/2 ≤ x ≤ π/2
  2. cos-1 (cos x) = x provided 0 ≤ x ≤ π
  3. tan-1 (tan x) = x provided -π/2 < x < π/2
  4. sin(sin-1 x) = x provided -1 ≤ x ≤ 1
  5. cos(cos-1 x) = x provided -1 ≤ x ≤ 1
  6. tan(tan-1 x) = x provided x ∈ R
  7. cosec(cosec-1 x) = x provided -1 ≤ x ≤ ∞ or -∞ < x ≤ 1
  8. sec(sec-1 x) = x provided 1 ≤ x ≤ ∞ or -∞ < x ≤ 1
  9. cot(cot-1 x) = x provided -∞ < x < ∞
  10. sin^{-1}(\frac{2x}{1 + x^2}) = 2 tan^{-1}x
  11. cos^{-1}(\frac{1 - x^2}{1 + x^2}) = 2 tan^{-1}x
  12. tan^{-1}(\frac{2x}{1 - x^2}) = 2 tan^{-1}x
  13. 2cos-1 x = cos-1 (2x2 - 1)
  14. 2sin-1x = sin-1 2x√(1 - x2)
  15. 3sin-1x = sin-1(3x - 4x3)
  16. 3cos-1 x = cos-1 (4x3 - 3x)
  17. 3tan-1x = tan-1((3x - x3/1 - 3x2))
  18. sin-1x + sin-1y = sin-1{ x√(1 - y2) + y√(1 - x2)}
  19. sin-1x - sin-1y = sin-1{ x√(1 - y2) - y√(1 - x2)}
  20. cos-1 x + cos-1 y = cos-1 [xy - √{(1 - x2)(1 - y2)}]
  21. cos-1 x - cos-1 y = cos-1 [xy + √{(1 - x2)(1 - y2)}
  22. tan-1 x + tan-1 y = tan-1(x + y/1 - xy)
  23. tan-1 x - tan-1 y = tan-1(x - y/1 + xy)
  24. tan-1 x + tan-1 y +tan-1 z = tan-1 (x + y + z - xyz)/(1 - xy - yz - zx)

➢Practice: Solved Examples

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