Tangent is Sine divided by Cosine

Theorem

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

Then:

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

where $\tan$, $\sin$ and $\cos$ mean tangent, 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:

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

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds \frac y x$

$\ds $

$=$

$\ds \tan \theta$

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

$\blacksquare$