If $z = cis(2k\pi/5)$, $z \neq 1$, then what is $(z+1/z)^2+(z^2 + 1/z^2)^2=$? question 20, part c in the picture:
I substituted the first time as $4 \cos^2(2k \pi/5)$ and the second term as $4 \cos^2(4k \pi/5)$, and then tried writing one term in terms of the other using the identity $\cos 2a = 2 \cos^2 a- 1$. I even tried bringing in $\sin$ but I didn't get anywhere. The answer is supposed to be $3$. Can someone solve it?


The sum is $4+z^2+z^{-2}+z^4+z^{-4}$.
Show that it equals $4+z+z^2+z^3+z^4$.
