Question

If $x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ and $x\sin \theta =y\cos \theta ,$then $(x^2+y^2)$ is equal to

• A. $2$
• B. $0$
• C. $3$
• D. $1$

Answer 1

The correct option is D $1$

Given,

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

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

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

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

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

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

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

$\therefore x=\cos \theta$

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

$\therefore y=\sin \theta$

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