Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$
Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$
$\frac{1+i}{1-i}-\frac{1-i}{1+i}=\frac{(1+i)^{2}-(1-i)^{2}}{(1-i)(1+i)}$
$=\frac{1+i^{2}+2 i-1-i^{2}+2 i}{1^{2}+1^{2}}$
$=\frac{4 i}{2}=2 i$
$\therefore\left|\frac{1+i}{1-i}-\frac{1-i}{1+i}\right|=|2 i|=\sqrt{2^{2}}=2$
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