If $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = .....\left| {{z_n}} \right| = 1,$$     then the value of $$\left| {{z_1} + {z_2} + ..... {z_n}} \right| - \left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + ..... + \frac{1}{{{z_n}}}} \right|$$         is,

Solution :
$$\eqalign{ & {z_1}{{\bar z}_1} = {z_2}{{\bar z}_2} = ..... = {z_n}{{\bar z}_n} = 1 \cr & \Rightarrow \,{{\bar z}_1} = \frac{1}{{{z_1}}},{{\bar z}_2} = \frac{1}{{{z_2}}},{{\bar z}_3} = \frac{1}{{{z_3}}},.....,{{\bar z}_n} = \frac{1}{{{z_n}}} \cr & \therefore \,\left| {{z_1} + {z_2} + ..... + {z_n}} \right| - \left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + ..... + \frac{1}{{{z_n}}}} \right| \cr & = \,\left| {{z_1} + {z_2} + ..... + {z_n}} \right| - \left| {{{\bar z}_1} + {{\bar z}_2} + ..... + {{\bar z}_n}} \right| = 0 \cr} $$