Prove that tanh² x + sech²x = 1

by RoRi

November 19, 2015

Prove the following identity,

$\tanh^2 x + \operatorname{sech}^2 x = 1.$

From the definitions of hyperbolic tangent and hyperbolic secant we have,

$\begin{align*} \tanh^2 x + \operatorname{sech}^2 x &= \frac{\sinh^2 x}{\cosh^2 x} + \frac{1}{\cosh^2 x} \\ &= \frac{\sinh^2 x + 1}{\cosh^2 x} \\ &= \frac{\cosh^2 x}{\cosh^2 x} \\ & = 1. \qquad \blacksquare \end{align*}$

Stumbling Robot

Prove that tanh² x + sech²x = 1

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