Cotangent of Straight Angle

Theorem

 * $\cot 180^\circ = \cot \pi$ is undefined

where $\cot$ denotes cotangent.

Proof
From Cotangent is Cosine divided by Sine:
 * $\cot \theta = \dfrac {\cos \theta} {\sin \theta}$

When $\sin \theta = 0$, $\dfrac {\cos \theta} {\sin \theta}$ can be defined only if $\cos \theta = 0$.

But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.

When $\theta = \pi$, $\sin \theta = 0$.

Thus $\cot \theta$ is undefined at this value.