complex numbers - Proof of Euler's formula that doesn't use differentiation?

So I saw a 'proof' of the sine and cosine angle addition formulae,
i.e. $\sin(x+y)=\sin x\cos y+\cos x \sin y$, using Euler's formula, $e^{ix}=\cos x+i\sin x$.
By multiplying by $e^{iy}$, you can get the desired result.

However, this 'proof' appears to be circular reasoning, as all proofs I have seen of Euler's formula involve finding the derivative of the sine and cosine functions. But to find the derivative of sine and cosine from first principles requires the use of the sine and cosine angle addition formulae.

So is there any proof of Euler's formula that doesn't involve finding the derivative of sine or cosine?
I know you can prove the trigonometric formulas geometrically, but it is more laborious to do.