Inverse Hyperbolic Cotangent is Odd Function/Proof 1

Theorem

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


Proof

\(\ds \map {\coth^{-1} } {-x}\) \(=\) \(\ds y\)
\(\ds \leadstoandfrom \ \ \) \(\ds -x\) \(=\) \(\ds \coth y\) Definition 1 of Inverse Hyperbolic Cotangent
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds -\coth y\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds \map \coth {-y}\) Hyperbolic Cotangent Function is Odd
\(\ds \leadstoandfrom \ \ \) \(\ds \coth^{-1} x\) \(=\) \(\ds -y\) Definition 1 of Inverse Hyperbolic Cotangent

$\blacksquare$

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