What is Euler's proof of his formula ${e^{ix}=\cos(x)+i\sin(x)}$

Question

I've read several proofs of the Euler's formula $$e^{ix}=\cos(x)+i\sin(x)$$ but I want to know how Euler's himself prove it at the first time, how did he think about it?

Answer 1

Wikipedia says: "Around 1740 Euler turned his attention to the exponential function instead of logarithms, and obtained the formula used today that is named after him. It was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions."

The proof using series is given here.