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Prove that tanh (-x) = – tanh x

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*}

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