1. Basic Functional Identities
These identities help convert between inverse trig functions and simplify expressions.
- \( \sin(\sin^{-1} x) = x \), for \(x \in [-1, 1]\)
- \( \cos(\cos^{-1} x) = x \), for \(x \in [-1, 1]\)
- \( \tan(\tan^{-1} x) = x \), for all real \(x\)
The inverses work perfectly only on their allowed domains.
2. Complementary Angle Identities
These identities relate inverse functions of sine, cosine and tangent to each other.
- \( \sin^{-1} x + \cos^{-1} x = \dfrac{\pi}{2} \)
- \( \tan^{-1} x + \cot^{-1} x = \dfrac{\pi}{2} \)
These are frequently used in simplifications and proofs.
3. Reciprocal Identities
Using the reciprocal relationships from trigonometry:
- \( \sec^{-1} x = \cos^{-1}\left(\dfrac{1}{x}\right) \), for \(|x| \ge 1\)
- \( \csc^{-1} x = \sin^{-1}\left(\dfrac{1}{x}\right) \), for \(|x| \ge 1\)
- \( \cot^{-1} x = \tan^{-1}\left(\dfrac{1}{x}\right) \), for \(x > 0\)
4. Odd–Even Identities
These identities help when expressions contain negative values.
- \( \sin^{-1}(-x) = -\sin^{-1}(x) \)
- \( \tan^{-1}(-x) = -\tan^{-1}(x) \)
- \( \cos^{-1}(-x) = \pi - \cos^{-1}(x) \)
- \( \cot^{-1}(-x) = \pi - \cot^{-1}(x) \)
5. Sum and Difference Identities for tan⁻¹
These are very useful in simplifying expressions involving sums of inverse tangents.
- \( \tan^{-1} a + \tan^{-1} b = \tan^{-1}\left(\dfrac{a + b}{1 - ab}\right) \), if \(ab < 1\)
- \( \tan^{-1} a - \tan^{-1} b = \tan^{-1}\left(\dfrac{a - b}{1 + ab}\right) \)
These identities come from the tangent addition and subtraction formulas.
6. Important Composed Identities
These identities help combine two different inverse trig functions into one.
- \( \sin^{-1} x = \tan^{-1}\left(\dfrac{x}{\sqrt{1 - x^2}}\right) \), for \(|x| < 1\)
- \( \cos^{-1} x = \tan^{-1}\left(\dfrac{\sqrt{1 - x^2}}{x}\right) \), for \(x > 0\)
- \( \tan^{-1} x = \sin^{-1}\left(\dfrac{x}{\sqrt{1 + x^2}}\right) \)
7. Double Angle–Type Identities
These mirror the structure of standard double-angle formulas.
- \( 2\sin^{-1} x = \sin^{-1}(2x\sqrt{1 - x^2}) \)
- \( 2\tan^{-1} x = \tan^{-1}\left(\dfrac{2x}{1 - x^2}\right) \), for \(x \ne \pm 1\)
8. Examples to Understand the Identities
Example 1: Simplify \(\sin^{-1}(x) + \cos^{-1}(x)\).
\(\sin^{-1}(x) + \cos^{-1}(x) = \dfrac{\pi}{2}\)
Example 2: Evaluate \(\tan^{-1}(1) + \tan^{-1}(2)\).
Let a = 1, b = 2: \( \tan^{-1}(1) + \tan^{-1}(2) = \tan^{-1}\left(\dfrac{1 + 2}{1 - 2}\right) = \tan^{-1}(-3) \).