Given :
$x\sin^3\theta+y\cos^3\theta=\sin\theta\cos\theta$------(1)
$x\sin\theta=y\cos\theta$
Let $x\sin\theta=y\cos\theta=k$-------(2)
We can write equation (1) as
$k(\sin^2\theta+\cos^2\theta)=\sin\theta\cos\theta)$
$k(1)=\sin\theta\cos\theta$
$\sin\theta\cos\theta=k$
From equation (2) we have
$x=\cos\theta,y=\sin\theta$
$x^2=\cos^2\theta,y^2=\sin^2\theta$
$\Rightarrow x^2+y^2=1$
Hence (c) is the correct option.