Question

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

Answer 1

Find the modulus:

Let the $z=\frac{(1+i)}{(1-i)}-\frac{(1-i)}{(1+i)}$

Solve,

$$\begin{gathered} \Rightarrow z=\frac{(1+i)}{(1-i)}-\frac{(1-i)}{(1+i)} \\ \Rightarrow z=\frac{{(1+i)}^2-{(1-i)}^2}{1^2-{\left(i\right)}^2}\{i=\sqrt{-1}\} \\ \Rightarrow z=\frac{1+{\left(i\right)}^2+2i-1-{\left(i\right)}^2+2i}{1-{\left(\sqrt{-1}\right)}^2} \\ \Rightarrow z=\frac{2i+2i}{1+1} \\ \Rightarrow z=\frac{4i}{2}=2i \end{gathered}$$

We know that, modulus of $z=x+iy$ is $\sqrt{x^2+y^2}$

So, for $z=0+2i$ modulus will be,

$$\begin{gathered} =\sqrt{0^2+{\left(2\right)}^2} \\ \Rightarrow \sqrt{{\left(2\right)}^2} \\ =\sqrt{4} \\ =2 \end{gathered}$$

Therefore, the modulus is $2$.