Can someone point me to a good, rigorous proof of Euler's formula (e^(ix) = cos(x) + i*sin(x)), starting from the definitions of sin(x) and cos(x) using the triangle (sin(x) = Opposite / Hypotenuse, cos(x) = Adjacent / Hypotenuse) and the definition of e^x as either Bernoulli's number (from the limit) or as the non-trivial self-differentiating function, and then proving everything from there? Pythagoras' formula can be assumed. Basic common-knowledge trig identities can be assumed. But these are only because I already know how to derive them! I want to be able to write a rigorous proof. Before someone proposes the Taylor series proof: fine. I can prove it using the Taylor series. But that is not how I wish to define sin(x) and cos(x) or e^x, so if you want to do it like that, you need to first give me a rigorous proof of Taylor's theorem from first-principles! I suspect there are easier ways of proving Euler's formula?