I am proving the formula $$ \sin{ \left(x+y \right)} =\sin{x} \cos{y}+\cos{x} \sin{y}$$ by using Euler's formula. This sum formula is needed when proving the derivative of sine.
I am only wondering if then I make a circular argument, since all proofs of Euler's formula which I have seen require knowing the derivatives of sine and cosine.
Ahlfors in his complex analysis book defines $e^z$ to satisfy the equation $f'(z)=f(z)$ for all $z$ in the complex plane, and $f(0)=1$. From this definition he derives the power series expansion of $e^z$ and show it converges everywhere. Also He shows $e^{a+b}=e^ae^b$ using Leibniz rule for derivatives. Then defines $\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$ and $\cos(z)=\frac{e^{iz}+e^{-iz}}{2}$. From this Eulers formula is derivable without the derivative of $\sin$.