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  • The number of solution of the equation <img alt="\mathrm{\cos \left(x+\frac{\pi}{3}\right) \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4} \cos ^{2} 2 x}" src="https://entrancecorner.oncodecogs.

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

Option: 1

8


Option: 2

5


Option: 3

6


Option: 4

7


Answers (1)

best_answer

\begin{gathered} \mathrm {\cos \left(x+\frac{\pi}{3}\right) \quad \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4} \cos ^{2} 2 x }\\ \mathrm{\cos ^{2} \frac{\pi}{3}-\sin ^{2} x=\frac{1-2 \sin ^{2} x}{4} }\\ \mathrm{\frac{1}{4}-\sin ^{2} x=\frac{1}{4}-\frac{\sin ^{2} x}{2} }\\ \mathrm{\sin ^{2} x=0} \\ \mathrm{\sin x=0 \quad \therefore \quad x=-3 \pi,-2 \pi,-\pi, 0,} \\ \mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ \pi, 2 \pi, 3 \pi} \end{gathered}

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