Question

If $\left|z\right|=1$ prove that $\frac{1+z}{1+z}=z.$

Answer 1

Given, $\left|z\right|=1\Rightarrow {\left|z\right|}^2=1\Rightarrow z\overline{z}=1$

Now, $\frac{1+z}{1+\overline{z}}=\frac{z\overline{z}+z}{1+\overline{z}}=\frac{z(1+\overline{z})}{(1+\overline{z})}=z[\because z\overline{z}=1]$

$1+\overline{z}$