Sine of sum proof

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.

• No, it just requires that $e^{i s} e^{it} = e^{i(s+t)}$. Aug 30, 2020 at 19:38
• It really depends on your starting point. Rudin ("Real & complex analysis") defines $\exp, \cos, \sin$ in terms of their power series and derives relationships from there. Aug 30, 2020 at 19:40

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 $$\sin$$.