# 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.

## Proof

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

Then:

 $\displaystyle \frac {\cos \theta} {\sin \theta}$ $=$ $\displaystyle \frac {x / r} {y / r}$ Cosine of Angle in Cartesian Plane and Sine of Angle in Cartesian Plane $\displaystyle$ $=$ $\displaystyle \frac x r \frac r y$ $\displaystyle$ $=$ $\displaystyle \frac x y$ $\displaystyle$ $=$ $\displaystyle \cot \theta$ Cotangent of Angle in Cartesian Plane

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

$\blacksquare$