Question

Find the number of solutions of the equations;

${\sin }^{3}x\cos x+{\sin }^{2}x.{\cos }^{2}x+\sin x.{\cos }^{3}x=1$,when $xϵ[0,2\pi ]$

Answer 1

${\sin }^{3}x\cos x+{\sin }^{2}x{\cos }^{2}x+\sin x{\cos }^{3}x=1$

$\Rightarrow \sin x\cos x\left({\sin }^{2}x+{\cos }^{2}x\right)+{\sin }^{2}x{\cos }^{2}x=1$

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

$\Rightarrow \frac{{\sin }^{2}2x}{4}+\frac{\sin 2x}{2}-1=0$

$\Rightarrow {\sin }^{2}2x+2\sin 2x-4=0$

$\Rightarrow \sin 2x=\frac{-2\pm \sqrt{4+16}}{2}=\frac{-2\pm \sqrt{20}}{2}=\frac{-2\pm 2\sqrt{5}}{2}$

$=-1\pm \sqrt{5}$

As $-1\le \sin \theta \le 1$

Hence no solution