Explanation

(c)
$\sec{x\cos{5x = - 1}}$
$\Rightarrow \ \cos{5x = - \cos x}$
$\Rightarrow \ 5x = 2n\pi \pm (\pi - x)$
$\Rightarrow \ x = \frac{(2n + 1)\pi}{6}\ or\ \frac{(2n - 1)\pi}{6}$
The possible values of $x$ which lies in the interval $(0,\ 2\pi)$ are $\frac{\pi}{4},\frac{\pi}{6},\frac{\pi}{2},\frac{3\pi}{4},\frac{5\pi}{6},\frac{5\pi}{4},\frac{7\pi}{6},\frac{7\pi}{4},\frac{9\pi}{6}\ and\frac{11\pi}{6}$