The question is divided into 3 parts, part a), b) and c). I understood a) and b) just fine, but it the last question im struggling with. 

Here we go:
Let $Q=\cos(2\pi/5)+i\sin(2\pi/5)$

a) show $Q$ is a root of $z^5 - 1 = 0$ 

b) Show that $(Q^4 + Q^3 + Q^2 + Q + 1)(Q-1) = Q^5 -1$ 
Then deduce that
$Q^4 + Q^3 + Q^2 + Q + 1 = 0$ 

c) Hence show that $\cos\frac{2\pi}{5} + \cos\frac{4\pi}{5} = -\frac{1}{2}$

So here i write out $Q^4 + Q^3 + Q^2 + Q + 1$ using 
$Q = \cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5}$

which is correct according to the solution, however, the next step throws me off so badly. 

This: 

$$1 + \cos \frac{2\pi}{5} + i\sin \frac{2\pi}{5} + \cos\frac{4\pi}{5} + i\sin\frac{4\pi}{5} + \cos \frac{6\pi}{5} + i\sin \frac{6\pi}{5} + \cos \frac{8\pi}{5} + i\sin \frac{8\pi}{5}$$

Turns into this: 

$$1 + \cos\frac{2\pi}{5} + i\sin \frac{2\pi}{5} + \cos \frac{4\pi}{5} + i\sin \frac{4\pi}{5} + \cos \frac{4\pi}{5} - i\sin \frac{4\pi}{5} + \cos \frac{2\pi}{5} - i\sin \frac{2\pi}{5}$$

So could someone be so kind to explain to me what exactly is going on between these two steps? Because im totally clueless :/ Everything up to this point, and what comes after is all fine and dandy, but its the last two stars that i dont get. Why do the sings change when the angles are replaced? Thanks =)