Question

Solve: $\cos 2x\cos 4x\cos 6x$.

Answer 1

$\cos 2x\cos 4x\cos 6x$

$\frac{1}{2}\left[2\cos 2x\cos 4x\right]\cos 6x$

$\frac{1}{2}\cos 6x[\cos 6x+\cos 2x](2\cos A\cos B=\cos (A+B)+\cos (A-B\left)\right)$

$\frac{1}{2}[{\cos }^{2}6x+\cos 6x\cos 2x]$

$\frac{1}{2}[\frac{1}{2}+\frac{\cos 12x}{2}+\frac{2\cos 6x\cos 2x}{2}]$ $[\cos 2\theta =2{\cos }^2\theta -1]$

$\frac{1}{2}[\frac{1}{2}+\frac{\cos 12x}{2}+\frac{\cos 8x}{2}+\frac{\cos 4x}{2}]$

$\frac{1}{4}[1+\cos 12x+\cos 8x+\cos 4x]$

$\therefore \cos 2x\cos 4x\cos 6x=\frac{1}{4}[1+\cos 12x+\cos 8x+\cos 4x]$.