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  4. 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 π] is

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Q. 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

Complex Numbers and Quadratic Equations

Solution:

We have $\sin ^3 x \cos x+\sin ^2 x \cos ^2 x+\sin x \cos ^3 x=1 \Rightarrow \sin x \cos x\left(\sin ^2 x+\sin x \cos x+\cos ^2 x\right)=1$
$\Rightarrow \frac{\sin 2 x}{2}\left(1+\frac{1}{2} \sin 2 x\right)=1 \Rightarrow \sin 2 x(2+\sin 2 x)=4 \Rightarrow \sin ^2 2 x+2 \sin 2 x-4=0 $
$\therefore \sin 2 x=\frac{-2 \pm \sqrt{4+16}}{2}=-1 \pm \sqrt{5} \text { (Not possible) }$
As, $-1 \leq \sin 2 x \leq 1 \forall x \in R$
Hence, number of solutions equals 0 .