Prove that cosh (2x) = cosh²x + sinh² x

by RoRi

November 17, 2015

Prove that

$\cosh (2x) = \cosh^2 x + \sinh^2 x.$

Proof. We know from this previous exercise (Section 6.19, Exercise #6) that

$\cosh (x +y) = \cosh x \cosh y + \sinh x \sinh y.$

Therefore,

$\begin{align*} \cosh(2x) &= \cosh (x+x) \\ &= \cosh x \cosh x + \sinh x \sinh x \\ &= \cosh^2 x + \sinh^2 x. \qquad \blacksquare \end{align*}$

Stumbling Robot