If $$\left| z \right| = 1$$  then $$\frac{{1 + z}}{{1 + \overline z }}$$  is equal to

Solution :
$$\eqalign{ & \frac{{1 + z}}{{1 + \overline z }} = \frac{{z\overline z + z}}{{1 + \overline z }}\left( {\because \,\,\left| z \right| = 1\, \Rightarrow \,{{\left| z \right|}^2} = 1\,\, \Rightarrow \,z\overline z = 1} \right) \cr & \frac{{1 + z}}{{1 + \overline z }} = \frac{{z\left( {\overline z + 1} \right)}}{{1 + \overline z }} = z. \cr} $$