Cotangent Function is Periodic on Reals

Theorem
The cotangent function is periodic on the set of real numbers $$\mathbb{R}$$ with period $\pi$.

Proof
From the definition of the cotangent function, we have that $$\cot x = \frac {\cos x} {\sin x}$$.

We have:

$$ $$ $$

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

From Nature of Sine Function, we have that $$\sin \ > 0$$ on the interval $$\left({0 \, . \, . \, \pi}\right)$$.

From Derivative of Monotone Function, $$\cot x$$ is strictly decreasing on that interval, and hence can not have a period of less than $$\pi$$.

Hence the result.