If $z$ is a complex number of unit modulus and argument theta If $z$ is a complex number such that $|z|=1$ and $\text{arg} z=\theta$, then what is  $$\text{arg}\frac{1 + z}{1+ \overline{z}}?$$

$$ \frac{1+z}{1 + \bar z} = \frac{1+z}{1 + \bar z} \times \frac{1+z}{1 + z} = \frac{(1+z)^2}{|1+z|^2} $$
the argument should be $ 2 \arg (1 +z) = 2 \arctan \left( y \over x+1\right)$
Let $x = \cos \theta$ and $y = \sin \theta $, we have $\arctan \left( \frac{y}{x+1}\right) = \arctan \left( \frac{\sin \theta}{\cos \theta+1}\right) = \arctan \left( \frac{\sin \frac{\theta}{2}}{\cos {\theta\over 2 }}\right)$
$2 \arg (1+z)^2 = \arg (z)$
