# 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:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \tan \left({x + \pi}\right)$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {\sin \left({x + \pi}\right)} {\cos \left({x + \pi}\right)}$$ $$\displaystyle$$ $$\displaystyle$$ from Sine and Cosine are Periodic on Reals $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {-\sin x} {-\cos x}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \tan x$$ $$\displaystyle$$ $$\displaystyle$$

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 Shape 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.

$\blacksquare$