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Prove that tanh2 x + sech2x = 1

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*}

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