Question

If $|z|=1$ then value of $|\overline{z}+z^{-1}{|}^2$ is

Answer 1

$[\because |z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2+2Re(z_1\overline{z_2}\left)\right]$
$\therefore |\overline{z}+z^{-1}{|}^2=|\overline{z}{|}^2+|z^{-1}{|}^2+2Re\left(\overline{z}(\overline{z^{-1})}\right)$
$[\because |z|=|\overline{z}|]$
$=|z{|}^2+{\left|\frac{1}{z}\right|}^2+2Re\left(1\right)$
$=1+1+2\left(1\right)[\because |z|=1]$
$=4$