Hyperbolic Tangent Function is Odd

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Theorem

Let $\tanh: \C \to \C$ be the hyperbolic tangent function on the set of complex numbers.


Then $\tanh$ is odd:

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


Proof

\(\ds \map \tanh {-x}\) \(=\) \(\ds \frac {\map \sinh {-x} } {\map \cosh {-x} }\) Definition 2 of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \frac {-\sinh x} {\map \cosh {-x} }\) Hyperbolic Sine Function is Odd
\(\ds \) \(=\) \(\ds \frac {-\sinh x} {\cosh x}\) Hyperbolic Cosine Function is Even
\(\ds \) \(=\) \(\ds -\tanh x\) Definition 2 of Hyperbolic Tangent

$\blacksquare$


Also see


Sources