Tangent Function is Periodic on Reals

Theorem
The tangent function is periodic on the set of real numbers $\R$ with period $\pi$.

This can be written:
 * $\tan x = \map \tan {x \bmod \pi}$

where $x \bmod \pi$ denotes the modulo operation.

Proof
From Derivative of Tangent Function, we have that:
 * $\map {D_x} {\tan x} = \dfrac 1 {\cos^2 x}$

provided $\cos x \ne 0$.

From Shape of Cosine Function, we have that $\cos > 0$ on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From Derivative of Monotone Function, $\tan x$ is strictly increasing on that interval, and hence can not have a period of less than $\pi$.

Hence the result.