Explanation

(b)
$\sqrt{\sin^{2}{x - \sin{x + \frac{1}{2}}}} = \sqrt{\left( \sin x - \frac{1}{2} \right)^{2} + \frac{1}{4}} \geq \frac{1}{2},\ \forall x$
and $\sec^{2}{y \geq 1,\ \forall y,\ so\ 2^{\sec^{2}y} \geq 2}$. Hence, the above inequality holds only for those values of $x$ and $y$ for which $\sin{x = \frac{1}{2}}$ and $\sec^{2}{y = 1}$. Hence, $x = \frac{\pi}{6},\frac{5\pi}{6},\frac{13\pi}{6},\frac{17\pi}{6}$ and $y = 0,\ \pi,\ 2\pi,\ 3\pi$. Hence, required number of ordered pairs are 16