The locus of the point  $\left(\mathrm{cosec} \mathrm{\theta }+\cot \mathrm{\theta }, \mathrm{cosec} \mathrm{\theta }-\cot \mathrm{\theta } \right)$ is

 Let P=(x, y) be locus of point.

Given that $$\begin{gathered} Given (\mathrm{x}, \mathrm{y})=(\mathrm{cosec\theta }+\mathrm{cot\theta }, \mathrm{cosec\theta }-\mathrm{cot\theta }) \\ \Rightarrow \mathrm{x}= \mathrm{cosec\theta }+\mathrm{cot\theta } - \left(1\right) \\ \mathrm{y}= \mathrm{cosec\theta }+\mathrm{cot\theta } - \left(2\right) \\ \left(1\right)\times \left(2\right) \Rightarrow \mathrm{xy}=(\mathrm{cosec\theta }+\mathrm{cot\theta }) (\mathrm{cosec\theta }-\mathrm{cot\theta }) \\ \Rightarrow \mathrm{xy}={\mathrm{cosec}}^2\mathrm{\theta }-{\cot }^2\mathrm{\theta } \\ \Rightarrow \mathrm{xy}=1 \end{gathered}$$