Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$? How can we find the RHS if we don't know what it is? (instead of proving the identity itself)
I could find a geometric solution in Wikipedia, but is there any solution that doesnt require drawing something?
Edit: I now saw some nice proofs using $e$ and Euler's identity. I would appreciate anything NOT using them too, as a change.
Out of curiosity, Is there a proof that says you actually "can't" prove the identity using only other simple 1-variable identities like $\sin^2\alpha+\cos^2\alpha=1$?

If you don't want to use geometry, you can do it with complex powers of $e$:
$$
\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}\\
\cos(x)=\frac{e^{ix}+e^{-ix}}{2}\\
\sin(x+y)=\frac{e^{i(x+y)}-e^{-i(x+y)}}{2i}
$$
You can now also express the right hand side of your equality in terms of powers of $e$, and this wil give the same result.
