Question

Value it :-
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

Given-

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

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

$\to \left(x\sin \theta \right)\left({\sin }^2\theta \right)+y{\cos }^3\theta =\sin \theta \cos \theta$

$\to y\cos \theta \left({\sin }^2\theta \right)+y{\cos }^3\theta =\sin \theta \cos \theta$

$\to y\cos \theta ({\sin }^2\theta +{\cos }^2\theta )=\sin \theta \cos \theta$

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

$y=\sin \theta$

similarly,

$\to x=\cos \theta$

so,

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

Hence Proved.