Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ and $x\sin \theta =y\cos \theta$ Equation (i) may be written as $x\sin \theta \cdot {\sin }^2\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ $\Rightarrow y\cos \theta {\sin }^2\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ $\Rightarrow y\cos \theta \left({\sin }^2\theta +{\cos }^2\theta \right)=\sin \theta \cos \theta$ $\Rightarrow y\cos \theta =\sin \theta \cos \theta \therefore y=\sin \theta \dots$ Putting the value of y from (iii) in (ii), we get $\mathrm{x}\sin \theta =\sin \theta \cdot \cos \theta \Rightarrow \mathrm{x}=\cos \theta \dots$ (iv) Squaring (iii) and (iv) and adding, we get ${\mathrm{x}}^2+{\mathrm{y}}^2={\cos }^2\theta +{\sin }^2\theta =1$