Prove that \(4\cos x\cos\left(\frac{\pi}{3}+x\right)\cos\left(\frac{\pi}{3}-x\right)=\cos3x\)
Solution
\[
\begin{aligned}
&4\cos x\cos\left(\frac{\pi}{3}+x\right)\cos\left(\frac{\pi}{3}-x\right)\\[4pt]
&=2\cos x\left[2\cos\left(\frac{\pi}{3}+x\right)\cos\left(\frac{\pi}{3}-x\right)\right]\\[4pt]
&=2\cos x\left[\cos\frac{2\pi}{3}+\cos2x\right]\\[4pt]
&=2\cos x\left[-\frac12+\cos2x\right]\\[4pt]
&=-\cos x+2\cos x\cos2x\\[4pt]
&=-\cos x+\cos3x+\cos x\\[4pt]
&=\cos3x
\end{aligned}
\]