Question

If $x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ and $x\sin \theta =y\cos \theta$, prove that $x^2+y^2=1$.

Answer 1

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

$\left(x\sin \theta \right){\sin }^2\theta +\left(y\cos \theta \right){\cos }^2\theta =\sin \theta \cos \theta$

$\left(x\sin \theta \right){\sin }^2\theta +\left(x\sin \theta \right){\cos }^2\theta =\sin \theta \cos \theta$

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

$x=\cos \theta$

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

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

$y=\sin \theta$

Therefore,

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