If $\mathrm{pcosec}\mathrm{\theta }+\mathrm{qcot}\mathrm{\theta }=2\text{ and }{\mathrm{p}}^2{\mathrm{cosec}}^2\mathrm{\theta }-{\mathrm{q}}^2{\cot }^2\mathrm{\theta }=5$ then the value of $81{\mathrm{p}}^{-2}-{\mathrm{q}}^{-2}$ is

$\text{ We have }p\mathrm{cosec}\theta +q\cot \theta =2$

$\text{ And }(p\mathrm{cosec}\theta {)}^2-(q\cot \theta {)}^2=5$

$\Rightarrow p\mathrm{cosec}\theta -q\cot \theta =\frac{5}{2}$

From (1) and (2

$\begin{array}{l}\mathrm{pcosec}\mathrm{\theta }=\frac{9}{4},\mathrm{qcot}\mathrm{\theta }=\frac{-1}{4}\\ \text{ Now }{\mathrm{cosec}}^2\mathrm{\theta }-{\cot }^2\mathrm{\theta }=1\\ \Rightarrow \frac{81}{16{\mathrm{p}}^2}-\frac{1}{16{\mathrm{q}}^2}=1\\ \Rightarrow 81{\mathrm{p}}^{-2}-{\mathrm{q}}^{-2}=16\\ \Rightarrow \sqrt{81{\mathrm{p}}^{-2}-{\mathrm{q}}^{-2}}=4\end{array}$