Inverse Hyperbolic Cosecant is Odd Function

Theorem

Let $x \in \R$.

Then:

$\map {\csch^{-1} } {-x} = -\csch^{-1} x$

where $\map {\csch^{-1} } {-x}$ denotes the inverse hyperbolic cosecant function.

Proof

\(\ds \map {\csch^{-1} } {-x}\)

\(=\)

\(\ds y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds -x\)

\(=\)

\(\ds \csch y\)

Definition 1 of Inverse Hyperbolic Cosecant

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds -\csch y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds \map \csch {-y}\)

Hyperbolic Cosecant Function is Odd

\(\ds \leadstoandfrom \ \ \)

\(\ds \csch ^{-1} x\)

\(=\)

\(\ds -y\)

Definition 1 of Inverse Hyperbolic Cosecant

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.67$: Relations Between Inverse Hyperbolic Functions