# Inverse Tangent of i

## Theorem

The inverse tangent of $i$ is not defined.

## Proof

Aiming for a contradiction, suppose $\tan z_0 = i$.

 $\displaystyle \dfrac {\sin z_0} {\cos z_0}$ $=$ $\displaystyle i$ Definition of Tangent Function $\displaystyle \leadsto \ \$ $\displaystyle \sin z_0$ $=$ $\displaystyle i \cos z_0$ $\displaystyle \leadsto \ \$ $\displaystyle \sin^2 z_0$ $=$ $\displaystyle -\cos^2 z_0$ $\displaystyle \leadsto \ \$ $\displaystyle \sin^2 z_0 + \cos^2 z_0$ $=$ $\displaystyle 0$

This contradicts the theorem Sum of Squares of Sine and Cosine:

$\sin^2 z_0 + \cos^2 z_0 = 1$

Hence the result by Proof by Contradiction.

$\blacksquare$