Summary

Principal value branches: Each inverse trigonometric function has a restricted range (principal value branch) to make it single-valued. For example, $\sin^{-1} x$ has range $[-\frac{\pi}{2}, \frac{\pi}{2}]$, $\cos^{-1} x$ has $[0, \pi]$, $\tan^{-1} x$ has $(-\frac{\pi}{2}, \frac{\pi}{2})$, etc.

Domain and range: $\sin^{-1} x$ is defined for $x \in [-1, 1]$, $\cos^{-1} x$ for $x \in [-1, 1]$, $\tan^{-1} x$ for $x \in \mathbb{R}$, $\cot^{-1} x$ for $x \in \mathbb{R}$, $\sec^{-1} x$ for $|x| \ge 1$, $\csc^{-1} x$ for $|x| \ge 1$.

Inverse property: For $x$ in the domain, $\sin(\sin^{-1} x) = x$ and $\sin^{-1}(\sin y) = y$ only if $y$ is in the principal branch of $\sin^{-1}$. Same holds for other functions.

Conversion formulas: $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$, $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$, $\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}$ for all $x$ in their respective domains.

Negative arguments: $\sin^{-1}(-x) = -\sin^{-1} x$, $\cos^{-1}(-x) = \pi - \cos^{-1} x$, $\tan^{-1}(-x) = -\tan^{-1} x$, $\cot^{-1}(-x) = \pi - \cot^{-1} x$.

Sum and difference formulas: $\tan^{-1} x + \tan^{-1} y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$ (with appropriate adjustments for $xy > 1$ or $x,y$ signs), and $\tan^{-1} x - \tan^{-1} y = \tan^{-1}\left(\frac{x-y}{1+xy}\right)$. …