Question

If $\left|z_1\right|=\left|z_2\right|=....=\left|z_n\right|=1$, prove that
$[z_1+z_2+....+z_n]=|\frac{1}{z_1}+\frac{1}{z_2}+....+\frac{1}{z_n}|$.

Answer 1

Given:

$|z_1|=|z_2|=|z_3|=.....=|z_n|=1$

Now,

$|z_1+z_2+z_3+.....+z_n|=|\frac{z_1\overline{z_1}}{\overline{z_1}}+\frac{z_2\overline{z_2}}{\overline{z_2}}+\frac{z_3\overline{z_3}}{\overline{z_3}}+....+\frac{z_n\overline{z_n}}{\overline{z_n}}|$

$=|\frac{|z_1{|}^2}{\overline{z_1}}+\frac{|z_2{|}^2}{\overline{z_2}}+\frac{|z_3{|}^2}{\overline{z_3}}+....+\frac{|z_n{|}^2}{\overline{z_n}}|$

$=|\frac{1}{\overline{z_1}}+\frac{1}{\overline{z_2}}+\frac{1}{\overline{z_3}}+....+\frac{1}{\overline{z_n}}|$

$=\overline{|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}+....+\frac{1}{z_n}|}$

$=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}+....+\frac{1}{z_n}|$

Hence, $LHS=RHS$.