Question

The number of solutions of the equation $\cos \left(x+\frac{\pi}{3}\right) \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4} \cos ^{2} 2 x, x \in[-3 \pi, 3 \pi]$ is :

• A. 8
• B. 5
• C. 6
• D. 7

Answer 1

Answer: (D)

$\cos \left(\frac{\pi}{3}+x\right) \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4} \cos ^{2} 2 x$
$x \in[-3 \pi, 3 \pi]$
$4\left(\cos ^{2}\left(\frac{\pi}{3}\right)-\sin ^{2} x\right)=\cos ^{2} 2 x$
$4\left(\frac{1}{4}-\sin ^{2} x\right)=\cos ^{2} 2 x$
$1-4 \sin ^{2} x=\cos ^{2} 2 x$
$1-2(1-\cos 2 x)=\cos ^{2} 2 x$
let $\cos 2 x=t$
$-1+2 \cos 2 x=\cos ^{2} 2 x$
$t^{2}-2 t+1=0$
$(t-1)^{2}=0$
$t=1 \cos 2 x=1$
$2 x=2 n \pi$
$x=n \pi$
$n=-3,-2,-1,0,1,2,3$