Is Euler's formula valid for complex arguments

Question

I found this question here :

Evaluate $\cos(z)$, given that $z = i \log(2+\sqrt{3})$

It says that - $$e^{-iz} = \cos(z) - i \sin(z)$$ isn't necessarily true because $$\sin z$$ is imaginary (for the particular value of z in the link). But, the proof of Euler's formula using Maclaurin series suggests that it should be valid for complex arguments too. What am I missing?

Answer 1

No, it doesn't say that $$e^{iz}=\cos(iz)-i\sin(iz)$$is not necessarily true! It is true. What it says in that other answer is that $\cos(iz)$ is not necessarily the real part of $e^{iz}$.

Answer 2

You are missing nothing: for every complex number $z$, $e^{iz}=\cos z+i\sin z$.