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Prove that coth2 x – csch2 x = 1

by RoRi

Prove the following identity,

    \[ \operatorname{coth}^2 x - \operatorname{csch}^2 x = 1. \]

Proof. From the definitions of hyperbolic cotangent and hyperbolic cosecant we have,

    \begin{align*} \operatorname{coth}^2 x - \operatorname{csch}^2 x &= \frac{\cosh^2 x}{\sinh^2 x} - \frac{1}{\sinh^2 x} \\ &= \frac{\cosh^2 x - 1}{\sinh^2 x} \\ &= \frac{\sinh^2 x}{\sinh^2 x} \\ &= 1. \qquad \blacksquare \end{align*}

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