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 $\pi$.


Proof

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

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