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?