Question

\[ \frac{\left(1+\sin\frac{4\pi}{9}-i\cos\frac{4\pi}{9}\right)^6} {\left(1+\sin\frac{4\pi}{9}+i\cos\frac{4\pi}{9}\right)^6} = \ ? \]

• A. $i$
• B. $\frac{\sqrt3-i}{2}$
• C. $\frac{1-\sqrt3\,i}{2}$
• D. $1+i$

Hint:

Whenever a quotient of conjugates appears, convert it into exponential form \[ \frac{re^{i\theta}}{re^{-i\theta}} = e^{2i\theta}. \] Then powers become very easy to evaluate.

Answer 1

The Correct Option is C