Answer
The identity has been verified
Work Step by Step
$$\eqalign{
& {\text{We have the identity }}\cosh \left( {x + y} \right) = \cosh x\cosh y + \sinh x\sinh y \cr
& {\text{Now}}{\text{, if we set }}y = x{\text{ in the given identity}},{\text{ then }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\cosh \left( {x + x} \right) = \cosh x\cosh x + \sinh x\sinh x \cr
& {\text{simplifying}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\cosh \left( {2x} \right) = {\cosh ^2}x + {\sinh ^2}x \cr} $$
