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

Just a nice point in general, I found for you in my notes. If $z=\frac{x_1+iy_1}{x_2+iy_2}$, then you can try to write $z$ as $a+ib$ for some proper $a,b$. The way is a bit similar to @Mula's answer. We get, $$z=a+ib,~~ a=\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2},~~ b=\frac{-x_1y_2+y_1x_2}{x_2^2+y_2^2}$$ Now, I think, you can find the answer by this another way also. Try it!
