Question

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

Answer 1

$$\begin{gathered} (1-i{)}^n{\left(1-\frac{1}{i}\right)}^n={\left(1-i\right)}^n{\left(1-\frac{i^4}{i}\right)}^n[\because i^4=1] \\ ={\left(1-i\right)}^n{\left(1-i^3\right)}^n \\ ={\left(1-i\right)}^n{\left(1+i\right)}^n[\because i^3=-i] \\ ={\left[(1-i)(1+i)\right]}^n \\ =(1-i^2{)}^n \\ =2^n[\because i^2=-1] \end{gathered}$$

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