Question

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

Answer 1

Let $z=\frac{1+i}{1-i}-\frac{1-i}{1+i}=\frac{{(1+i)}^2-{(1-i)}^2}{(1-i)(1+i)}$
$=\frac{1+i^2+2i-1-i^2+2i}{1^2+1^2}=\frac{4i}{2}=2i$
$\therefore \left|z\right|=\left|2i\right|=\sqrt{2^2}=2$