Question

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

Answer 1

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

$=\frac{1+i^2+2i-1-i^2+2i}{1^2+1^2}$

$=\frac{4i}{2}=2i$

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