Cotangent Function is Periodic on Reals

Theorem
The real cotangent function is periodic with period $\pi$.

Proof
Also, from Derivative of Cotangent Function:
 * $\map {D_x} {\cot x} = -\dfrac 1 {\sin^2 x}$

provided $\sin x \ne 0$.

From Shape of Sine Function, $\sin$ is strictly positive on the interval $\openint 0 \pi$.

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

Hence the result.