Inverse Tangent is Odd Function

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In particular: Expand for $\tan^{-1}$ on complex plane, include this as a corollary
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Theorem

Everywhere that the function is defined:

$\map \arctan {-x} = -\arctan x$

Proof

$\ds \map \arctan {-x}$

$=$

$\ds y$

$\ds \leadstoandfrom \ \ $

$\ds -x$

$=$

$\ds \tan y:$

$\ds -\frac \pi 2 \le y \le \frac \pi 2$

Definition of Real Arctangent

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

$\ds -\tan y:$

$\ds -\frac \pi 2 \le y \le \frac \pi 2$

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

$\ds \map \tan {-y}:$

$\ds -\frac \pi 2 \le y \le \frac \pi 2$

Tangent Function is Odd

$\ds \leadstoandfrom \ \ $

$\ds \arctan x$

$=$

$\ds -y$

Definition of Real Arctangent

$\blacksquare$

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.82$