If $$x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta  and  x \sin \theta - y \cos \theta = 0$$, then the value of $$x^2 + y^2$$ equals

$$x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta $$ --> eq 1

$$ x \sin \theta - y \cos \theta = 0$$

$$ x \sin \theta = y \cos \theta$$ --->eq2

substituting in eq1

$$y\cos \theta \sin^2 \theta + y \cos^3 \theta = \sin \theta \cos \theta $$

taking $$ y\cos \theta  common$$

$$y\cos \theta (\sin^2 \theta +  \cos^2 \theta )= \sin \theta \cos \theta $$ { we know $$\sin^2 \theta +  \cos^2 \theta  = 1$$}

$$y\cos \theta= \sin \theta \cos \theta $$ 

$$y= \sin \theta$$

substituting in eq 2 

$$ x \sin \theta = \sin \theta \cos \theta$$ 

x = $$\cos \theta$$

 $$x^2 + y^2$$ =  $$\sin^2\theta + \cos^2\theta$$ = 1