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