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  $$\text{arg}\frac{1 + z}{1+ \overline{z}}?$$

Multiplying both numerator and denominator by $z$, we get:
$$\arg\left(\frac{1+z}{1+\bar{z}}\right)=\arg\left(\frac{z+z^{2}}{z+1}\right)=\arg\left(\frac{z(1+z)}{1+z}\right)=\arg\left(z\right)$$
We are told that $\arg(z)=\theta$, therefore:
$$\arg\left(\frac{1+z}{1+\bar{z}}\right)=\theta$$
