If z is a unimodular complex number, then arg  $\left({\mathrm{z}}^2+\overline{\mathrm{z}}\right)=$

z in a unimodular $\iff \left|\mathrm{Z}\right|=1$
Let $\mathrm{z}=\cos \mathrm{\theta }+\mathrm{isin}\mathrm{\theta }$, then  $\overline{\mathrm{z}}=\cos \mathrm{\theta }-\mathrm{isin}\mathrm{\theta }$
$\begin{array}{l}\arg \left({\mathrm{z}}^2+\overline{\mathrm{z}}\right)=\arg (\cos 2\mathrm{\theta }+\mathrm{isin}2\mathrm{\theta }+\cos \mathrm{\theta }-\mathrm{isin}\mathrm{\theta })={\tan }^{-1}\left(\frac{\sin 2\mathrm{\theta }-\sin \mathrm{\theta }}{\cos 2\mathrm{\theta }+\cos \mathrm{\theta }}\right)\\ ={\tan }^{-1}\left[\tan \frac{\mathrm{\theta }}{2}\right]=\frac{\mathrm{\theta }}{2}=\frac{1}{2}\arg \mathrm{z}\end{array}$