Question:
ifÂ
$\left(\frac{1-i}{1+i}\right)^{100}=a+i b ;$ then find $(a, b)$
Solution:
According to the question,
We have,
$a+i b=\left(\frac{1-i}{1+i}\right)^{100}$
$=\left[\frac{1-i}{1+i} \cdot \frac{1-i}{1-i}\right]^{100}$
$=\left[\frac{(1-i)^{2}}{1-i^{2}}\right]^{100}$
$=\left(\frac{1-2 \mathrm{i}+\mathrm{i}^{2}}{1+1}\right)^{100}$
$=\left(-\frac{2 \mathrm{i}}{2}\right)^{100}$
$=\left(i^{4}\right)^{25}$
= 1
Hence, (a, b) = (1, 0)
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.