Question

The sum of all solutions of the equation

$\cos x.\cos\left( \frac{\pi}{3} + x \right).\cos\left( \frac{\pi}{3} - x \right) = \frac{1}{4},x \in \lbrack 0,6\pi\rbrack$ is

• A. $\sin(\theta/3) = 6\pi$
• B. $30\pi$
• C. $\frac{110\pi}{3}$
• D. None of these

Answer 1

Answer: (B)

$30\pi$

Answer 2

Here, $\cos x\left( \frac{1}{4}\cos^{2}x - \frac{3}{4}\sin^{2}x \right) = \frac{1}{4}$ or

$\frac{\cos x}{4}\left( 4\cos^{2}x - 3 \right) = \frac{1}{4}$ or $\cos 3x = 1$

$\Rightarrow 3x = 2n\pi$ $\Rightarrow x = \frac{2n\pi}{3},$ where $n = 0,1,2,3,4,5,6,7,8,9$;

$\therefore$The required sum $= \frac{2\pi}{3}\sum_{n = 0}^{9}n$= 30π.