Inverse Hyperbolic Cotangent is Odd Function/Proof 1
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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 |
$\blacksquare$