Explanation

(c)
$\left( 1 + i\sqrt{3} \right)^{n} + \left( 1 - i\sqrt{3} \right)^{n}$
$= 2^{n}\left\lbrack \left( \frac{1}{2} + \frac{i\sqrt{3}}{2} \right)^{n} + \left( \frac{1}{2} - \frac{i\sqrt{3}}{2} \right)^{n} \right\rbrack$
$= 2^{n}\left\lbrack \left( \cos{\frac{\pi}{3} + i\sin\frac{\pi}{3}} \right)^{n} + \left( \cos{\frac{\pi}{3} - i\sin\frac{\pi}{3}} \right)^{n} \right\rbrack$
$= 2^{n}\left\lbrack 2\cos\frac{n\pi}{3} \right\rbrack = 2^{n + 1}\cos\frac{n\pi}{3}$