• # question_answer The number of solutions of the equation ${{\sin }^{3}}x\cos x+{{\sin }^{2}}x{{\cos }^{2}}x+\sin x{{\cos }^{3}}x=1,$ in the interval $[0,2\pi ],$ is A) 4 B) 2 C) 1 D) 0

The given equation can be written as $\sin x\cos x[{{\sin }^{2}}x+\sin x\cos x+{{\cos }^{2}}x]=1$ or $\sin x\cos x\left[ 1+\sin x\cos x \right]=1$ or  $\sin 2x\,[2+{{\sin }^{2}}x]=4$$\Rightarrow$$\sin 2x=\frac{-2\pm \sqrt{4+16}}{2}=-1\pm \sqrt{5}$Which is not possible.