Question

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

Answer 1

$$\begin{gathered} \frac{1+i}{1-i}-\frac{1-i}{1+i} \\ =\frac{\left(1+i\right)\left(1+i\right)-\left(1-i\right)\left(1-i\right)}{\left(1-i\right)\left(1+i\right)} \\ =\frac{1+i^2+2i-1-i^2+2i}{1^2-i^2} \\ =\frac{4i}{2}\left(\because i^2=-1\right) \\ =2i \\ \therefore \left|2i\right|=\sqrt{0^2+2^2} \\ =2\left(\because \left|a+bi\right|=\sqrt{a^2+b^2}\right) \\ \Rightarrow \left|\frac{1+i}{1-i}-\frac{1-i}{1+i}\right|=2 \end{gathered}$$