Back at the university we have proven (lot of work) that if $$S(X)C(Y)+C(X)S(Y) = S(X+Y)$$ and $$C(X)C(Y)-S(X)S(Y) = C(X+Y)$$ then $S(X)$ is $\sin(x)$ and $C(X)$ is $\cos(x)$ (or constant $0$, meh). What is this theorem called..?
Later note by someone other than the original poster:
An amazingly large number of people, in posted answers and comments (some now deleted) have MISSED THE POINT. These are not the angle addition formulas for the sine and the cosine. In those formulas, one assumes the function are the sine and cosine and shows that these equations hold. In this problem, it's the other way around: One assumes these equations hold and then proves, rather than assuming from the outset, that the functions are the sine and cosine. I even rejected an edit to the original posting that would have written $\sin$ and $\cos$ in place of $S$ and $C$. That would have made the question incomprehensible!
Please: stop doing this. --- Michael Hardy