Properties of Inverse Trigonometric Functions

Inverse trigonometric functions undo their respective trigonometric functions, but only within the principal value branches.

• \( \sin(\sin^{-1} x) = x \) for \(x \in [-1,1]\)
• \( \cos(\cos^{-1} x) = x \) for \(x \in [-1,1]\)
• \( \tan(\tan^{-1} x) = x \) for all real x

However:

• \( \sin^{-1}(\sin x) \neq x \) in general
• \( \cos^{-1}(\cos x) \neq x \) in general

because the output must lie in the principal value interval.

These properties help simplify expressions involving negative inputs.

• \( \sin^{-1}(-x) = -\sin^{-1}(x) \) (odd function behaviour)
• \( \tan^{-1}(-x) = -\tan^{-1}(x) \)
• \( \cos^{-1}(-x) = \pi - \cos^{-1}(x) \) (reflection property)
• \( \cot^{-1}(-x) = \pi - \cot^{-1}(x) \)

Using the reciprocal relations of sec, cosec and cot:

• \( \sec^{-1} x = \cos^{-1} \left(\dfrac{1}{x}\right) \), for \(|x| \geq 1\)
• \( \csc^{-1} x = \sin^{-1} \left(\dfrac{1}{x}\right) \), for \(|x| \geq 1\)
• \( \cot^{-1} x = \tan^{-1} \left(\dfrac{1}{x}\right) \), for positive x

These properties connect inverse sine and inverse cosine, inverse tangent and inverse cotangent.

• \( \sin^{-1} x + \cos^{-1} x = \dfrac{\pi}{2} \)
• \( \tan^{-1} x + \cot^{-1} x = \dfrac{\pi}{2} \)

These are useful for simplifying expressions involving sums or differences:

• \( \tan^{-1} a + \tan^{-1} b = \tan^{-1}\left(\dfrac{a + b}{1 - ab}\right) \) if \(ab < 1\)
• \( \tan^{-1} a - \tan^{-1} b = \tan^{-1}\left(\dfrac{a - b}{1 + ab}\right) \)

Inverse functions also follow double-angle-like transformations.

• \( 2\sin^{-1} x = \sin^{-1}(2x\sqrt{1 - x^2}) \)
• \( 2\tan^{-1} x = \tan^{-1}\left(\dfrac{2x}{1 - x^2}\right) \)

Example 1: Simplify \(\sin^{-1}(-\dfrac{1}{2})\).

Example 2: Evaluate \(\cos^{-1}(-\dfrac{1}{2})\).