Question

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

Answer 1

To find $\to$ Modules of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$

$=|\frac{1+i}{1-i}-\frac{1-i}{1+i}|=\left|\frac{(1+i{)}^2-(1-i{)}^2}{(1-i)(1+i)}\right|$

$=\left|\frac{1+i^2+2i-1-i^2+2i}{1+i-i-i^2}\right|=\left|\frac{4i}{1-i^2}\right|$

$=\left|\frac{4i}{2}\right|=\left|2i\right|$

$[\therefore \sqrt{-1}$ and $i^2=-1]$