Inverse Cosecant is Odd Function

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

Everywhere that the function is defined:

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

Proof

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

$=$

$\ds y$

$\ds \leadstoandfrom \ \ $

$\ds -x$

$=$

$\ds \csc y:$

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

Definition of Arccosecant

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

$\ds -\csc y:$

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

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

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

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

Cosecant Function is Odd

$\ds \leadstoandfrom \ \ $

$\ds \arccsc x$

$=$

$\ds -y$

Definition of Arccosecant

$\blacksquare$

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.85$