Explanation

[Solution by Dr. Abubacker] With $z=e^{i\theta}$: $\frac z{\bar z}+\frac{\bar z}z=2\cos2\theta$, so $|2\cos2\theta|=1$, i.e. $\cos2\theta=\pm\frac12$. In $[0,2\pi)$ each sign gives $4$ values of $\theta$ ($2\theta$ in $[0,4\pi)$), for a total of $8$. Answer (2).