The Tangent: Derivative, Monotonicity, Bijectivity onto $\mathbb{R}$ and Special Values

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $\tan:(-\pi/2,\pi/2)\to\mathbb{R}$ be the tangent function, with $\sin$, $\cos$ and $\pi$ as there.

(i) Derivative. $\tan$ is differentiable, and its derivative $\tan'$ satisfies, for every $x\in(-\pi/2,\pi/2)$,

$$\tan'(x)=1+\tan^{2}x=\frac{1}{\cos^{2}x}.$$

(ii) Monotonicity and continuity. $\tan$ is strictly increasing, that is, $\tan x<\tan y$ for all $x,y\in(-\pi/2,\pi/2)$ with $x<y$, and $\tan$ is continuous on $(-\pi/2,\pi/2)$.

(iii) Bijectivity. $\tan:(-\pi/2,\pi/2)\to\mathbb{R}$ is bijective.

(iv) Special values.

$$\tan 0=0,\qquad \tan\frac{\pi}{4}=1 .$$