Question

If $\left(\frac{1+i}{1-i}\right)^{x}=1,$ then

• A. $x = 4n$, where $n$ is any positive integer
• B. $x = 2n$, where $n$ is any positive integer
• C. $x = 4n+1$, where $n$ is any positive integer
• D. $x = 2n+1$, where $n$ is any positive integer

Answer 1

Answer: (A)

$x = 4n$, where $n$ is any positive integer

Answer 2

$\frac{1+i}{1-i}=\frac{\left(1+i\right)^{2}}{2}=i$ $\left(\frac{1+i}{1-i}\right)^{x}=i^{x}$ $\Rightarrow x = 4n.$ Hence, (A) is the correct answer.