Prove that 2 sinh² (x/2) = cosh x – 1

by RoRi

November 19, 2015

Prove the following identity:

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

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

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

Stumbling Robot

Prove that 2 sinh² (x/2) = cosh x – 1

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