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Prove that 2 sinh2 (x/2) = cosh x – 1

Prove the following identity:

    \[ 2 \sinh^2 \frac{1}{2}x = \cosh x - 1. \]


Proof. We compute directly from the definition of \sinh x,

    \begin{align*}  2 \sinh^2 \frac{x}{2} &= 2 \left( \frac{e^{\frac{x}{2}} - e^{-\frac{x}{2}}}{2} \right)^2 \\  &= 2 \left( \frac{e^x - 2 + e^{-x}}{4} \right) \\  &= \frac{e^x + e^{-x}}{2} - 1 \\  &= \cosh x - 1. \qquad \blacksquare \end{align*}

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