Tangent Function is Periodic on Reals

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

Proof
From the definition of the tangent function, we have that $\tan x = \dfrac {\sin x} {\cos x}$.

We have:

Also, from Derivative of Tangent Function, we have that $\displaystyle D_x \left({\tan x}\right) = \frac 1 {\cos^2 x}$, provided $\cos x \ne 0$.

From Nature of Cosine Function, we have that $\cos \ > 0$ on the interval $\displaystyle \left({-\frac \pi 2 \, . \, . \, \frac \pi 2}\right)$.

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.