Explanation

(b)
Given, $\tan{\theta = \frac{1}{\sqrt{7}}}\ \ \ \Rightarrow \ \ \cot{\theta = \sqrt{7}}$
Now, $\frac{(cosec^{2}\theta - \sec^{2}{\theta)}}{(cosec^{2} + \sec^{2}{\theta)}} = \frac{(1 + \cot^{2}\theta - 1 - \tan^{2}\theta}{1 + \cot^{2}{\theta + 1 + \tan^{2}\theta}}$
$= \frac{\cot^{2}{\theta - \tan^{2}\theta}}{2 + \cot^{2}{\theta + \tan^{2}\theta}}$
$= \frac{\left( \sqrt{7} \right)^{2} - \left( \frac{1}{\sqrt{7}} \right)^{2}}{2 + \left( \sqrt{7} \right)^{2} + \left( \frac{1}{\sqrt{7}} \right)^{2}}$
$= \frac{49 - 1}{7} \times \frac{7}{63 + 1} = \frac{48}{64} = \frac{3}{4}$