Q. The value of $\left( 1 + \cos\frac{\pi}{6} \right)\left( 1 + \cos\frac{\pi}{3} \right)\left( 1 + \cos\frac{2\pi}{3} \right)\left( 1 + \cos\frac{7\pi}{6} \right)$ is
A) $\frac{3}{16}$
B) $\frac{3}{8}$
C) $\frac{3}{4}$
D) $\frac{1}{2}$
Explanation
(a)
$\left( 1 + \cos\frac{\pi}{6} \right)\left( 1 + \cos\frac{\pi}{3} \right)\left( 1 + \cos\frac{2\pi}{3} \right)\left( 1 + \cos\frac{7\pi}{6} \right)$
$= \left( 1 + \frac{\sqrt{3}}{2} \right)\left( 1 + \frac{1}{2} \right)\left( 1 - \frac{1}{2} \right)\left( 1 - \frac{\sqrt{3}}{2} \right)$
$= \left( 1 - \frac{3}{4} \right)\left( 1 - \frac{1}{4} \right) = \frac{1}{4} \times \frac{3}{4} = \frac{3}{16}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.