Question

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

Answer 1

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