# Inverse Tangent of i

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## 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.

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.6$. The Logarithm: Examples: $\text {(iv)}$