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:

$\tan x = \map \tan {x \bmod \pi}$

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


Proof

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

From Derivative of Tangent Function, we have that:

$\map {D_x} {\tan x} = \dfrac 1 {\cos^2 x}$

provided $\cos x \ne 0$.

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

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