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 $$\tan x = \frac {\sin x} {\cos x}$$.

We have:

$$ $$ $$

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

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

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.