complex numbers (Z - i / Z+i )^5 = 1

Question

I have attached a picture of the part I am stuck with. Can you please give some hints if not solve the problem?

I found

$$z = \tan(k(\pi)/5)$$ where $k = 1,2,3,4$.

Answer 1

$[(z+i)/(z-i)]^5 = 1 => z+i = e^{i 2k \pi/5} *(z-i)$

$z(1-e^{i 2k \pi/5}) = -i(1+e^{i 2k \pi/5})$

$z/-i = zi=(1+e^{i 2k \pi/5})/(1-e^{i 2k \pi/5})$

From part b, $zi=i \cot(k\pi/5) => z=1/\tan(k\pi/5)$, $k=1,2,3,4$.