Question

If $xsi{n}^3\theta +yco{s}^3\theta =\sin \theta \cos \theta andxsin\theta =y\cos \theta ,$ prove that
$x^2+y^2=xsec\theta$

Answer 1

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

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

$⟹x(\frac{y\cos \theta }{x}{)}^3+y{\cos }^3\theta =(\frac{y\cos \theta }{x})\cos \theta$

$⟹\frac{y^3}{x^2}{\cos }^3\theta +y{\cos }^3\theta =\frac{y}{x}{\cos }^2\theta$

$⟹\frac{y^3{\cos }^3\theta +x^{2}y{\cos }^3\theta }{x^2}=\frac{y}{x}{\cos }^2\theta$

$⟹\frac{y{\cos }^3\theta }{x^2}(x^2+y^2)=\frac{y}{x}{\cos }^2\theta$

$⟹x^2+y^2=x\text{sec}\theta$