Intuitive proof of the sin and cos addition formulas?

Question

I refer to these formulas:

$\sin(x±y)$ and $\cos(x±y)$

Is there an obvious or intuitive proof to derive their simpler identities in terms of $\sin(x), \sin(y), \cos(x), \cos(y)$?

I am tagging this with complex analysis because I'm open to such an explanation if it's simple enough to understand.

Answer 1

If you know complex analysis, it follows from $\exp(x+y)=\exp(x)\exp(y)$, by using $\exp(ix) = \cos x + i \sin x$ and comparing real and imaginary parts.