Explanation

(c)
$\left( \cos{\frac{\pi}{2} + i\sin\frac{\pi}{2}} \right)\left( \cos{\frac{\pi}{4} + i\sin\frac{\pi}{4}} \right)\left( \cos{\frac{\pi}{8} + \ i\sin\frac{\pi}{8}} \right)\ldots\infty$
$= \cos{\left( \ \frac{\pi}{2} + \frac{\pi}{4} + \frac{\pi}{8} + ...\infty \right) + i\sin\left( \ \frac{\pi}{2} + \frac{\pi}{4} + \frac{\pi}{8} + ...\infty \right)}$
$= \cos{\left( \frac{\ \frac{\pi}{2}}{1 - \frac{1}{2}} \right) + i\sin\left( \frac{\ \frac{\pi}{2}}{1 - \frac{1}{2}} \right)}$
$\cos{\pi + i\sin{\pi = - 1}}$