The values of x, between 0 and 2π , satisfying the equation $\cos 3 x + \cos 2 x = \sin \frac{3 x}{2} + \sin \frac{x}{2}$ are

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$\frac{13 π}{7}$

$\frac{9 π}{7}$

$\frac{5 π}{7}$

$\frac{π}{7}$

answer is A, B, C, D.

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Detailed Solution

$\cos 3 x + \cos 2 x = \sin \frac{3 x}{2} + \sin \frac{x}{2}$
$\begin{array}{l} 2 \cos \frac{5 x}{2} \cos \frac{x}{2} = 2 \sin xcos \frac{x}{2} \cos \frac{5 x}{2} \cos \frac{x}{2} - \sin xcos \frac{x}{2} = 0 \cos \frac{x}{2} (\cos \frac{5 x}{2} - \sin x) = 0 \\ ∴ \cos \frac{x}{2} = 0 \cos \frac{5 x}{2} = \sin x \\ \Rightarrow∴ \frac{x}{2} = 2 nπ \pm \frac{π}{2} or \frac{5 x}{2} = 2 nπ \pm (\frac{π}{2} - x), n \in Z \\ \Rightarrow x = \frac{π}{7}, \frac{5 π}{7}, π \frac{9 π}{7}, \frac{13 π}{7} \end{array}$