Question:
Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$
Solution:
$\frac{1+i}{1-i}-\frac{1-i}{1+i}=\frac{(1+i)^{2}-(1-i)^{2}}{(1-i)(1+i)}$
$=\frac{1+i^{2}+2 i-1-i^{2}+2 i}{1^{2}+1^{2}}$
$=\frac{4 i}{2}=2 i$
$\therefore\left|\frac{1+i}{1-i}-\frac{1-i}{1+i}\right|=|2 i|=\sqrt{2^{2}}=2$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.