Defining $\sin{x}$ and $\cos{x}$ via their power series, over the real numbers. With no allusion to complex numbers, or geometrical diagrams, how can one prove $$\forall x,y\in\mathbf{R},\\\sin{(x+y)} = \sin{x}\cos{y}+\cos{x}\sin{y} $$ starting from the barest bone theorems of power series and differentiable functions?
I have attempted to expand both sides via their power series representation and can see that the terms do look like they're matching, but a rigorous arguement feels out of reach. All I know is that the re-arrangement of terms on either side is valid since the series are absolutely convergent, but still, how to prove that the terms on both sides match for all higher order terms? (I've only checked the first and third order).
Apart from this method, is there a simpler way to prove it (once again, without complex numbers and starting only from the taylor series definitions). Similar to how one is able to prove $\sin^{2}{x}+\cos^{2}{x}=1$ via differentiating and then plugging in $x=0$. Is there a proof of the compound angle formula that is similar in spirit to the above?