Prove that $\displaystyle \frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\sec ^{2} \theta}=\frac{3}{4}$, if $\displaystyle \tan \theta=\frac{1}{\sqrt{7}}$

\[\begin{aligned} & \tan \theta=\frac{1}{\sqrt{7}} \\ & \Rightarrow \sec ^{2} \theta=\frac{8}{7} \text { and } \operatorname{cosec}^{2} \theta=8 \\ & \therefore \text { LHS }=\frac{8-\dfrac{8}{7}}{8+\dfrac{8}{7}}=\frac{3}{4}=\text { RHS } \end{aligned} \]