The number of solutions of the equation cos(x+ π 3) cos( π 3-x)= 1 4 cos^22x, x∈ [-3π ,3π ] is:

Given,

$$\mathrm{cos}(x+\frac{\pi }{3})\mathrm{cos}(x-\frac{\pi }{3})=\frac{1}{4}{\mathrm{cos}}^{2}(2x)$$cos(x+3π​)cos(x−3π​)=41​cos2(2x)

$$\Rightarrow 2\mathrm{cos}(x+\frac{\pi }{3})\mathrm{cos}(x-\frac{\pi }{3})=\frac{1}{2}{\mathrm{cos}}^{2}(2x)$$⇒2cos(x+3π​)cos(x−3π​)=21​cos2(2x)

$$\Rightarrow \mathrm{cos}(2x)+\mathrm{cos}(\frac{2\pi }{3})=\frac{1}{2}{\mathrm{cos}}^{2}2x$$⇒cos(2x)+cos(32π​)=21​cos22x

$$\Rightarrow \mathrm{cos}2x+(-\frac{1}{2})=\frac{1}{2}{\mathrm{cos}}^{2}2x$$⇒cos2x+(−21​)=21​cos22x

$$\Rightarrow {\mathrm{cos}}^{2}2x-2\mathrm{cos}2x+1=0$$⇒cos22x−2cos2x+1=0

$\Rightarrow {(\mathrm{cos}2x-1)}^{2}=0$⇒(cos2x−1)2=0or $\mathrm{cos}2x=1$cos2x=1

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

So, $x\in {-3\pi ,-2\pi ,-\pi ,0,\pi ,2\pi ,3\pi }$x∈−3π,−2π,−π,0,π,2π,3π

So total $7$7 solutions.