Question

Q. JEE MAIN 2022 (24 June) 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

Answer 1

4(cos²(π/3)−sin²x)=cos²2x

4(1/4−sin²x)=cos²2x

1−4sin²x=cos²2x

1−2(1−cos2x)=cos²2x

cos2⁡x = t

-1+ 2cos 2x =cos²2x

t²-2t+1=0

(t-1)²=0

t=1 \(\Rightarrow\) cos 2x=1

2x = 2nπ

x=nπ

n=−3,−2,−1,0,1,2,3

The number of solutions of the equation cos (x+π/3)cos(π/3-x)= 1/4cos²2x, x \(\in\) [-3π,3π] is 7