Explanation

(d)
$\ \sum_{k = 1}^{6}\left\lbrack \sin{\left( \frac{2k\pi}{7} \right) - i\cos\left( \frac{2k\pi}{7} \right)} \right\rbrack = - i\ \sum_{k = 1}^{6}\left( e^{\frac{2\pi i}{7}} \right)^{k}$
$= - i\left( r^{1} + r^{2} + ... + r^{6} \right)\ \ \ \left\lbrack let\ r = e^{\frac{2\pi i}{7}} \right\rbrack$
$= - ir\ \frac{(1 - r^{6})}{1 - r} = \frac{- i(r - r^{7})}{1 - r}$
$= \frac{- i(r - 1)}{1 - r} = i$ $\lbrack\because r^{7} = e^{2\pi i} = 1\rbrack$