If a point $(x,y)=(\tan \theta +\sin \theta ,\tan \theta -\sin \theta ),$ then the locus of (x, y) is

$\text{Let \hspace{0.17em}}(x,y)=(\tan \theta +\sin \theta ,\tan \theta -\sin \theta )\Rightarrow x=\tan \theta +\sin \theta ,y=\tan \theta -\sin \theta$

$xy=(\tan \theta +\sin \theta )(\tan \theta -\sin \theta )={\tan }^2\theta -{\sin }^2\theta ={\tan }^2\theta (1-{\cos }^2\theta )={\tan }^2\theta {\sin }^2\theta$

$x^2-y^2={\left(\tan \theta +\sin \theta \right)}^2-{(\tan \theta -\sin \theta )}^2=4\tan \theta .\sin \theta \Rightarrow {\left(x^2-y^2\right)}^2=16{\tan }^2\theta .{\sin }^2\theta$

$\Rightarrow$${(x^2-y^2)}^2=16xy$