Answer
$cosh~2x = cosh^2~x+sinh^2~x$
Work Step by Step
$cosh~2x = \frac{e^{2x}+e^{-2x}}{2}$
$cosh~2x = \frac{2e^{2x}+2e^{-2x}}{4}$
$cosh~2x = \frac{(e^{2x}+e^{-2x})+(e^{2x}+e^{-2x})}{4}$
$cosh~2x = \frac{(e^{2x}+2+e^{-2x})+(e^{2x}-2+e^{-2x})}{4}$
$cosh~2x = \frac{e^{2x}+2+e^{-2x}}{4}+\frac{e^{2x}-2+e^{-2x}}{4}$
$cosh~2x = (\frac{e^{x}+e^{-x}}{2})^2+(\frac{e^{x}-e^{-x}}{2})^2$
$cosh~2x = cosh^2~x+sinh^2~x$
