Cotangent is Cosine divided by Sine

Theorem

Let $\theta$ be an angle such that $\sin \theta \ne 0$.

Then:

$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$

where $\cot$, $\sin$ and $\cos$ mean cotangent, sine and cosine respectively.

Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.

Then:

$\ds \frac {\cos \theta} {\sin \theta}$

$=$

$\ds \frac {x / r} {y / r}$

$\ds $

$=$

$\ds \frac x r \frac r y$

$\ds $

$=$

$\ds \frac x y$

$\ds $

$=$

$\ds \cot \theta$

When $\sin \theta = 0$ the expression $\dfrac {\cos \theta} {\sin \theta}$ is not defined.

$\blacksquare$