Find the modulus of 1+i 1-i - 1-i 1+i - Vidyadip

Question

Find the modulus of $\frac{1+\text{i}}{1-\text{i}}-\frac{1-\text{i}}{1+\text{i}}$

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Answer

Let $\text{z}=\frac{1+\text{i}}{1-\text{i}}-\frac{1-\text{i}}{1+\text{i}}$
$=\frac{(1+\text{i})^2-(1-\text{i})^2}{(1-\text{i})(1+\text{i})}$
$=\frac{1^2+\text{i}^2+2\times1\times\text{i}-(1^2+\text{i}^2-2\times1\times\text{i})}{1^2+1^2}$
$=\frac{1-1+2\text{i}-(1-1-2\text{i})}{2}$
$=\frac{2\text{i}+2\text{i}}{2}$
$=\frac{4\text{i}}{2}$
$\Rightarrow\text{z}=2\text{i}$
$\therefore \ |\text{z}|=|2\text{i}|$
$=2|\text{i}| \ \Big(\because \ |\text{z}_1\text{z}_2|=|\text{z}_1|\times|\text{z}_2|\Big)$
$=2\times\text{i} \ \Big(\because \ |\text{i}|=1\Big)$
$=2$

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