The Inverse Tangent Function and its Properties

Tangent is not one-to-one over its full domain $\mathbb{R}\setminus\left\{\tfrac{\pi}2+k\pi:k\in\mathbb{Z}\right\}$, but $\tan:\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)\to\mathbb{R}$ IS a bijection.

Definition 4.5. For any real $x$, $\tan^{-1}x$ is the UNIQUE number $y\in\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$ such that $\tan y=x$. In symbols, $\tan^{-1}:\mathbb{R}\to\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$ is defined by $\tan^{-1}(x)=y\iff\tan y=x$ and $y\in\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$.

From the definition:

1. $y=\tan^{-1}x\iff x=\tan y$, for $x\in\mathbb{R}$ and $-\tfrac{\pi}2<y<\tfrac{\pi}2$.
2. $\tan(\tan^{-1}x)=x$ for EVERY real $x$ — unlike $\sin^{-1}$/$\cos^{-1}$, there is no domain restriction here, since $\tan^{-1}$'s own domain is already all of $\mathbb{R}$; $y=\tan^{-1}x$ is an odd function.
3. $\tan^{-1}(\tan x)=x$ if and only if $-\tfrac{\pi}2<x<\tfrac{\pi}2$. NOTE: $\tan^{-1}(\tan\pi)=0$, NOT $\pi$. Notes. $\tan:\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)\to\mathbb{R}$ and $\tan^{-1}:\mathbb{R}\to\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$. $\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$ is called the principal domain of tangent, and the values of $y=\tan^{-1}x$ are its principal values. …