Tangent of Right Angle

Theorem

 * $\tan 90^\circ = \tan \dfrac \pi 2$ is undefined

where $\tan$ denotes tangent.

Proof
From Tangent is Sine divided by Cosine, $\tan \theta = \dfrac {\sin \theta} {\cos \theta}$.

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

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

When $\theta = \dfrac \pi 2$, $\cos \theta = 0$.

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