Explanation

(c)
$\cos^{2}{\left( \frac{\pi}{3} - x \right) - \cos^{2}\left( \frac{\pi}{3} + x \right)}$
$= \left\lbrack \cos{\left( \frac{\pi}{3} - x \right) + \cos\left( \frac{\pi}{3} + x \right)} \right\rbrack\left\lbrack \cos{\left( \frac{\pi}{3} - x \right) - \cos\left( \frac{\pi}{3} + x \right)} \right\rbrack$
$= \left( 2\cos{\frac{\pi}{3}\cos x} \right)\left( 2\sin{\frac{\pi}{3}\sin x} \right)$
$= \sin{\frac{2\pi}{3}\sin{2x = \frac{\sqrt{3}}{2}\sin{2x}}}$
Hence, maximum value of given expression is$\ \frac{\sqrt{3}}{2}$