Question

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

Answer 1

Consider the given equation.

$x{\sin }^3\theta +y{\cos }^3\theta =\cos \theta .sin\theta$ …….. (1)

Since, $x\sin \theta =y\cos \theta$ ...... (2)

From equation (1), we get

$x{\sin }^3\theta +x\sin \theta {\cos }^2\theta =\cos \theta .sin\theta$

$x{\sin }^2\theta +x{\cos }^2\theta =\cos \theta$ ..... (3)

We know that,

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

Therefore, from equation (3), we have

$x=\cos \theta$

So, from equation (2), we have

$y=\sin \theta$

From equation (1), we get

$x{y}^3+y{x}^3=xy$

$xy(x^2+y^2)=xy$

$x^2+y^2=1$

Hence, proved.