Show that cosh(x+y) = cosh(x)cosh(y) + sinh(x)sinh(y)

RHS: cosh(x)cosh(y) + sinh(x)sinh(y) = 1/4(e^x + e^-x)(e^y + e^-y) + 1/4(e^x - e^-x)(e^y - e^-y) = 1/4(e^x.e^y + e^x.e^-y + e^-x.e^y + e^-x.e^-y + e^x.e^y - e^x.e^-y - e^-x.e^y + e^-x.e^-y) = 1/4(2e^x.e^y + 2e^-x.e^-y) = 1/2(e^x.e^y + e^-x.e^-y) = 1/2(e^(x+y) + e^-(x+y)) = cosh(x+y) [QED]