Question

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

Answer 1

\(\frac{1+i}{1+i}-\frac{1-i}{1+i}=\frac{(1+i)^2-(1-i)^2}{(1-i)(1+i)}\)

\(=\frac{1+i^2+2i-1-i^2+2i}{1^2+1^2}\)

\(=\frac{4i}{2}=2i\)

\(∴|\frac{1+i}{1-i}-\frac{1-i}{1+i}|=|2i|=\sqrt2^2=2\)