Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

The expression can be simplified using the identity $\sin \theta \sin (60^{\circ }-\theta )\sin (60^{\circ }+\theta )=\frac{1}{4}\sin (3\theta )$. Note that $\sin (x+\frac{2\pi }{3})=\sin (x+120^{\circ })=\sin (60^{\circ }-x)$. Thus, the given expression is $16\sin x\sin (x+60^{\circ })\sin (60^{\circ }-x)$. This simplifies to $16\left(\frac{1}{4}\sin (3x)\right)=4\sin (3x)$. The maximum value of $\sin (3x)$ is $1$. Therefore, the maximum value of $4\sin (3x)$ is $4\times 1=4$.