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Prove that 2 cosh2 (x/2) = cosh x + 1

by RoRi

Prove the following identity,

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

Proof. Computing directly from the definition of the hyperbolic cosine,

    \begin{align*} 2 \cosh^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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