Question

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

• A. $\frac{\pi }{2}-\theta$
• B. $\theta$
• C. $\pi -\theta$
• D. $-\theta$

Answer 1

The correct option is B $\theta$

Given: $|z|=1\Rightarrow z\overline{z}=1$

Now,

$\frac{1+z}{1+\overline{z}}$ $=\frac{1+z}{1+\frac{1}{z}}$

$=z.$

Hence, the argument is $\theta$