If $z$ is a complex number of unit modulus and argument  $\theta$, then $\arg \left(\frac{1+z}{1+\overline{z}}\right)$ equals 

: Note that  $\frac{1+z}{1+\overline{\overline{z}}}=\frac{1+z}{1+\frac{1}{z}}=z$

Observe that $\left|z^2\right|=1=z\overline{z}$

Then the arg of the number $\frac{1+z}{1+\overline{\overline{z}}}$ is just the argument

of $z$ and that's $\theta$.