If cot θ=3–√, find the value of cosec2 θ+cot2 θcosec2 θ−sec2 θ.

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 $\frac{21}{8}$.