Is the complex cosine function surjective?

Question

Let $\cos z=\frac{e^{iz} - e^{-iz}}{2}$ be the complex cosine function.

Then is $\cos:\mathbb{C}\rightarrow \mathbb{C}$ surjective?

If so, how do i prove this?

Answer 1

Hint: Do it the usual way, take an arbitrary element $w\in \mathbb C$ and try to find $z$ such that $\dfrac{e^{iz} + e^{-iz}}{2}=w$. To do this transform this equation in a polynomial of second degree on the variable $u$ with $u=e^{iz}$.