Prove that tanh (-x) = – tanh x

by RoRi

November 17, 2015

Prove that

$\tanh (-x) = - \tanh x.$

Proof. From the previous two exercises (here and here) we know

$\sinh (-x) = - \sinh x \qquad \text{and} \qquad \cosh (-x) = \cosh x.$

Therefore,

$\begin{align*} \tanh(-x) &= \frac{\sinh(-x)}{\cosh(x)} \\ &= -\frac{\sinh x}{\cosh x} \\ &= - \tanh x. \qquad \blacksquare \end{align*}$

Stumbling Robot