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Prove that cosh x – sinh x = e-x

Prove that

    \[ \cosh x - \sin x = e^{-x}. \]


Proof. Using the definition of the hyperbolic functions in terms of the exponential, we have

    \begin{align*}  \cosh x - \sinh x &= \frac{e^x+e^{-x}}{2} - \frac{e^x - e^{-x}}{2} \\  &= \frac{e^x + e^{-x} - e^x + e^{-x}}{2} \\  &= \frac{2e^{-x}}{2} \\  &= e^{-x}. \qquad \blacksquare \end{align*}

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