Question

If $(\frac{1+i}{1-i})^x=1$ then

• A. x = 4n + 1; n ∈ N
• B. x = 2n + 1, n ∈ N
• C. x = 2n; n ∈ N
• D. x = 4n; n ∈ N

Answer 1

The Correct Option is D

The given equation is $\left(\frac{1+i}{1-i}\right)^x = 1$. We can simplify the fraction $\frac{1+i}{1-i}$ by multiplying the numerator and denominator by $1+i$ to get: $$\frac{1+i}{1-i} = \frac{(1+i)^2}{(1-i)(1+i)} = \frac{1 + 2i - 1}{1 + 1} = \frac{2i}{2} = i$$ Therefore, the equation becomes: $$i^x = 1$$ The powers of $i$ cycle every 4 terms as $i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$.
Thus, for $i^x = 1$, $x$ must be a multiple of 4, i.e., $x = 4n$, where $n$ is a natural number.

The correct answer is (D) : x = 4n; n ∈ N