Inverse Trigonometry Sheet 2
Self-Adjusting (Composition) Properties
1. Form: $$f^{-1}(f(x)) = x$$
Valid when $$x$$ is in the Principal Value Range of the inverse function.
| Property | Validity Domain ($$x$$) |
|---|---|
| $$\sin^{-1}(\sin x) = x$$ | $$x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$$ |
| $$\cos^{-1}(\cos x) = x$$ | $$x \in [0, \pi]$$ |
| $$\tan^{-1}(\tan x) = x$$ | $$x \in (-\frac{\pi}{2}, \frac{\pi}{2})$$ |
| $$\text{cosec}^{-1}(\text{cosec } x) = x$$ | $$x \in [-\frac{\pi}{2}, \frac{\pi}{2}], x \neq 0$$ |
| $$\sec^{-1}(\sec x) = x$$ | $$x \in [0, \pi], x \neq \frac{\pi}{2}$$ |
| $$\cot^{-1}(\cot x) = x$$ | $$x \in (0, \pi)$$ |
2. Form: $$f(f^{-1}(x)) = x$$
Valid when $$x$$ is in the Domain of the inverse function.
| Property | Validity Domain ($$x$$) |
|---|---|
| $$\sin(\sin^{-1} x) = x$$ | $$x \in [-1, 1]$$ |
| $$\cos(\cos^{-1} x) = x$$ | $$x \in [-1, 1]$$ |
| $$\tan(\tan^{-1} x) = x$$ | $$x \in \mathbb{R}$$ |
| $$\text{cosec}(\text{cosec}^{-1} x) = x$$ | $$|x| \geq 1$$ |
| $$\sec(\sec^{-1} x) = x$$ | $$|x| \geq 1$$ |
| $$\cot(\cot^{-1} x) = x$$ | $$x \in \mathbb{R}$$ |