Question

If $x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ and $x\sin \theta =y\cos \theta$. Find the value of $x^2+y^2$.

Answer 1

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

$x=y\frac{\cos \theta }{\sin \theta }$

$\to x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$

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

$\to y=\sin \theta$

$\to x=y\frac{\cos \theta }{\sin \theta }=\cos \theta$

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