$Ifx{\sin }^3\theta +y{\cos }^3\theta =\cos \theta \sin \theta andx\sin \theta =y\cos \theta thenbye\lim nating\theta$

$\begin{array}{l}x{\sin }^3\theta +y{\cos }^3\theta =\cos \theta \sin \theta \\ \Rightarrow x{\left(\frac{y\cos \theta }{x}\right)}^3+y{\cos }^3\theta =\cos \theta \left(\frac{y\cos \theta }{x}\right)\Rightarrow \frac{y^3+x^{2}y}{x^2}{\cos }^3\theta =\frac{y{\cos }^2\theta }{x}\Rightarrow \cos \theta =\frac{x}{x^2+y^2}.\\ Similarly\sin \theta =\frac{y}{x^2+y^2}.{\cos }^2\theta +{\sin }^2\theta =1\Rightarrow \frac{x^2}{(x^2+y^2)}+\frac{y^2}{(x^2+y^2)}=1\Rightarrow x^2+y^2=1.\end{array}$