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. [Tex]sin^{-1}(\frac{2x}{1 + x^2}) = 2 tan^{-1}x[/Tex]
11. [Tex]cos^{-1}(\frac{1 – x^2}{1 + x^2}) = 2 tan^{-1}x[/Tex]
12. [Tex]tan^{-1}(\frac{2x}{1 – x^2}) = 2 tan^{-1}x[/Tex]
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)

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Inverse Trigonometric Identities: In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inverse trigonometric functions are generally used in fields like geometry, engineering, etc. The representation of inverse trigonometric functions are:

If a = f(b), then the inverse function is

 b = f^-1(a) 

Examples of inverse inverse trigonometric functions are sin^-1x, cos^-1x, tan^-1x, etc.