Question

The number of solutions of the equation $(\sin 3x+\sin x)\sin x+(\cos 3x-\cos x)\cos x=0$ in $[-\pi ,\pi ]$ is

• A. $0$
• B. $2$
• C. more than $2$ but finite
• D. infinitely many

Answer 1

The correct option is D infinitely many

$(\sin 3x+\sin x)\sin x+(\cos 3x-\cos x)\cos x=0\Rightarrow \sin 3x\sin x+\cos 3x\cos x+({\sin }^{2}x-{\cos }^{2}x)=0\Rightarrow \cos (3x-x)-\left(\cos 2x\right)=0\Rightarrow \cos 2x-\cos 2x=0$
Which is an identity, so it can have infinitely many solutions possible.