Solution for a complexed equation

Question

Find $z$ for the equation $e^z + e^{-z} = 0$.

So $$e^z + e^{-z} = 0 \iff e^z = -e^{-z} \iff e^z = e^{\pi i - z} \iff z = \pi i -z + 2\pi ik$$

I understand all expect the $2\pi ik$. Can you explain why the equation is true for all $k\in \mathbb{Z}$?

Answer 1

If $z=a+bi$, then $e^z=e^a(cos(b)+isin(b))$

Since the cos- and sin- function have periodicity $2\pi$, you get what you want.