For a positive integer n, find the value of
Question:

For a positive integer $n$, find the value of $(1-i)^{n}\left(1-\frac{1}{i}\right)^{n}$.

Solution:

$(1-i)^{n}\left(1-\frac{1}{i}\right)^{n}=(1-i)^{n}\left(1-\frac{i^{4}}{i}\right)^{n}$   $\left[\because i^{4}=1\right]$

$=(1-i)^{n}\left(1-i^{3}\right)^{n}$

$=(1-i)^{n}(1+i)^{n}$   $\left[\because i^{3}=-i\right]$

$=[(1-i)(1+i)]^{n}$

$=\left(1-i^{2}\right)^{n}$

$=2^{n} \quad\left[\because i^{2}=-1\right]$

Thus, the value of $(1-i)^{n}\left(1-\frac{1}{i}\right)^{n}$ is $2^{n}$.