Question

Evaluate \[ \left(\frac{1-i}{1+i}\right)^{2022} + \left(\frac{1+i}{1-i}\right)^{2021} \]

• A. \(-i\)
• B. \(i\)
• C. \(i+1\)
• D. \(i-1\)

Hint:

Convert complex fractions to polar form before exponentiation; then use properties \(i^4=1\) and \((-i)^2=-1\) to simplify powers.

Answer 1

The Correct Option is D

Step 1: Reduce each fraction to a single power of i.
Multiplying top and bottom by the conjugate, \(\frac{1-i}{1+i}=\frac{(1-i)^2}{2}=\frac{-2i}{2}=-i\), and similarly \(\frac{1+i}{1-i}=i\).

Step 2: Reduce the exponents using the period-4 cycle of powers of i.
Since \(2022=4(505)+2\), \((-i)^{2022}=(-i)^2=-1\).
Since \(2021=4(505)+1\), \((i)^{2021}=i^1=i\).

Step 3: Add the results.
\[ (-i)^{2022}+(i)^{2021} = -1+i = i-1 \]
\[ \boxed{i-1} \]