Tangent Function is Periodic on Reals

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Theorem

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


This can be written:

$\tan x = \tan \left({x \bmod \pi}\right)$

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


Proof

\(\displaystyle \tan \left({x + \pi}\right)\) \(=\) \(\displaystyle \frac {\sin \left({x + \pi}\right)} {\cos \left({x + \pi}\right)}\) Definition of Real Tangent Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\sin x} {-\cos x}\) Sine and Cosine are Periodic on Reals
\(\displaystyle \) \(=\) \(\displaystyle \tan x\)

From Derivative of Tangent Function, we have that:

$D_x \left({\tan x}\right) = \dfrac 1 {\cos^2 x}$

provided $\cos x \ne 0$.

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

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.

$\blacksquare$


Sources