How do you prove that cosh (x-y) = (cosh x. cosh y) - (sinh x . sinh y)

right ----> left

cosh x = [ e^x + e^(-x) ] / 2, sinh x = [ e^x - e^(- x) ] / 2

thus cosh x cosh y = { e^x e^y + e^(-x) e^y + e^x e^(-y) + e^(-x) e^(-y) } / 4

and sinh x sinh y = { e^x e^y - e^(-x e^y - e^x e^(-y) + e^(-x) e^(-y) } / 4

subtract [ 1/ 2] [e^(-x+y) + e^( x - y ) ] = cosh (x - y )