Solution to the linear functional equition with 2 variables.

I got this problem to solve:

f, g are continious limited functions on R

find all solutions to this system

f(x+y) = f(x)*f(y)-g(x)*g(y)

g(x+y) = f(x)*g(y)+f(y)*g(x)

f(0) = 1

g(0) = 0

I only found 1 solution by guessing, and it's g(x) = sin(rx+2pi*n), f(x) = cos(rx+2pi*k), where r is real and n,k are integers

Can you please give me the advice how to get more complete solution to this problem or how to prove that there's no more such a function?