Question

The value of $\left[\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}\right]^{-4}$ is

• A. $-i\pi$
• B. $i\pi$
• C. $i$
• D. $-i$
• E. $\pi$

Hint:

Negative powers simply reverse the angle sign.

Answer 1

The Correct Option is D

Step 1: Apply De Moivre’s theorem. \[ (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) \]

Step 2: Substitute values. \[ = \cos\left(-\frac{4\pi}{8}\right) + i\sin\left(-\frac{4\pi}{8}\right) \] \[ = \cos\left(-\frac{\pi}{2}\right) + i\sin\left(-\frac{\pi}{2}\right) \]

Step 3: Evaluate. \[ \cos(-\tfrac{\pi}{2}) = 0, \sin(-\tfrac{\pi}{2}) = -1 \] \[ = -i \] \[ \boxed{-i} \]