f(x+y) = (1/2) (a^{x+y} + a^{-(x+y)})

f(x+y) = (1/2) (a^{x-y} + a^{-(x-y)})

so:

f(x+y) + f(x-y) = (1/2) [a^{x+y} + a^{-(x+y)} + a^{x-y} + a^{-(x-y)}]

........ = (1/2) [ a^x (a^y-a^{-y}) + a^{-x} (a^y-a^{-y})]

........ = f(y) [ a^x + a^{-x}] = 2 f(y) f(x) = 2 f(x) f(y)

RonL