If $\cot \theta=\sqrt{3}$, find the value of $\frac{\cos e c^{2} \theta+\cot ^{2} \theta}{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}$.
Given: $\cot \theta=\sqrt{3}$
We have to find the value of the expression $\frac{\operatorname{cosec}^{2} \theta+\cot ^{2} \theta}{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}$.
We know that,
$\cot \theta=\sqrt{3} \Rightarrow \cot ^{2} \theta=3$
$\operatorname{cosec}^{2} \theta=1+\cot ^{2} \theta=1+(\sqrt{3})^{2}=4$
$\sec ^{2} \theta=\frac{1}{\cos ^{2} \theta}=\frac{1}{1-\sin ^{2} \theta}=\frac{1}{1-\frac{1}{\operatorname{cosec}^{2} \theta}}=\frac{1}{1-\frac{1}{4}}=\frac{4}{3}$
Therefore,
$\frac{\operatorname{cosec}^{2} \theta+\cot ^{2} \theta}{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}=\frac{4+3}{4-\frac{4}{3}}$
$=\frac{21}{8}$
Hence, the value of the given expression is $\frac{21}{8}$.