If one knows that the solutions of $y''+ y=0$ are two functions $s(x)$ and $c(x)$, and we know that $s(0)=0$, $s'(0)=1$, $c(0)=1$, $c'(0)=0$, then how can one start to prove that $s(x+a)=s(x)c(a)+c(x)s(a)$? What is it that has to be observed in order for one to understand how to initiate the proof?
One is not supposed to use the actual trigonometric functions in this case. Here we assume that we know nothing about sin and cos.