Cotangent Function is Periodic on Reals

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

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

We have:

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