The number of solutions of the equation (x + (π)/(3)) ((π)/(3) - x)...

Given,

$\cos \left(x + \dfrac{\pi}{3}\right)\cos \left(x - \dfrac{\pi}{3}\right) = \dfrac{1}{4}\cos^{2}\left(2x\right)$

$⇒ 2\cos \left(x + \dfrac{\pi}{3}\right)\cos \left(x - \dfrac{\pi}{3}\right) = \dfrac{1}{2}\cos^{2}\left(2x\right)$

$⇒ \cos \left(2x\right) + \cos \left(\dfrac{2\pi}{3}\right) = \dfrac{1}{2}\cos^{2}2x$

$⇒ \cos 2x + \left(- \dfrac{1}{2}\right) = \dfrac{1}{2}\cos^{2}2x$

$⇒ \cos^{2}2x - 2\cos 2x + 1 = 0$

$⇒ {\left(\cos 2x - 1\right)}^{2} = 0$or $\cos 2x = 1$

$⇒ 2x = - 6\pi, - 4\pi, - 2\pi, 0, 2\pi, 4\pi, 6\pi$

So, $x \in \left\{- 3\pi, - 2\pi, - \pi, 0, \pi, 2\pi, 3\pi\right\}$

So total $7$ solutions.