Cotangent Function is Periodic on Reals

Theorem

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

Proof

\(\ds \map \cot {x + \pi}\)

\(=\)

\(\ds \frac {\map \cos {x + \pi} } {\map \sin {x + \pi} }\)

Definition of Cotangent

\(\ds \)

\(=\)

\(\ds \frac {-\cos x} {-\sin x}\)

Cosine of Angle plus Straight Angle, Sine of Angle plus Straight Angle

\(\ds \)

\(=\)

\(\ds \cot x\)

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.

$\blacksquare$