If $\left(\frac{1-i}{1+i}\right)^{100}=a+i b$, where, $a b \in R$ and $i=\sqrt{-1}$, then $(a, b)$ is equal to

Question

Aicharya · Accepted Answer

$\begin{aligned} & \left(\frac{1-i}{1+i}ight)^{100}=\left\{\frac{(1-i)(1-i)}{(1+i)(1-i)}ight\}^{100}=\left(\frac{1-1-2 i}{1+1}ight)^{100}=(-i)^{100}=1=a+i b \ & \Rightarrow a=1 	ext { and } b=0\end{aligned}$

If $\left(\frac{1-i}{1+i}\right)^{100}=a+i b$, where, $a b \in R$ and $i=\sqrt{-1}$, then $(a, b)$ is equal to

Solution