Inverse Tangent of i

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Theorem

The inverse tangent of $i$ is not defined.


Proof

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

\(\ds \dfrac {\sin z_0} {\cos z_0}\) \(=\) \(\ds i\) Definition of Tangent Function
\(\ds \leadsto \ \ \) \(\ds \sin z_0\) \(=\) \(\ds i \cos z_0\)
\(\ds \leadsto \ \ \) \(\ds \sin^2 z_0\) \(=\) \(\ds -\cos^2 z_0\)
\(\ds \leadsto \ \ \) \(\ds \sin^2 z_0 + \cos^2 z_0\) \(=\) \(\ds 0\)

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