Inverse Hyperbolic Tangent is Odd Function/Proof 1
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Theorem
- $\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$
Proof
\(\ds \map {\tanh^{-1} } {-x}\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds -x\) | \(=\) | \(\ds \tanh y\) | Definition 1 of Inverse Hyperbolic Tangent | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds -\tanh y\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \map \tanh {-y}\) | Hyperbolic Tangent Function is Odd | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \tanh^{-1} x\) | \(=\) | \(\ds -y\) | Definition 1 of Inverse Hyperbolic Tangent |
$\blacksquare$