Hyperbolic Tangent Function is Odd

Theorem
Let $x \in \C$ be a complex number.

Let $\tanh x$ be the hyperbolic tangent of $x$.

Then:
 * $\tanh \left({-x}\right) = -\tanh x$

That is, the hyperbolic tangent function is odd.

Proof
Recall the definition of the hyperbolic tangent function:


 * $ \displaystyle \tanh x = \frac { \sinh x } { \cosh x } $

Then,

Also see

 * Hyperbolic Sine Function is Odd
 * Hyperbolic Cosine Function is Even
 * Tangent Function is Odd