Q. The value of $\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}\cos\frac{14\pi}{15}$ is
A) $\frac{1}{16}$
B) $\frac{1}{8}$
C) $\frac{1}{12}$
D) $\frac{1}{4}$
Explanation
(a)
Given, $\cos{24{^\circ}\cos{48{^\circ}\cos{96{^\circ}\cos{168{^\circ}}}}}$
$= - \cos{12{^\circ}\cos{24{^\circ}\cos{48{^\circ}\cos{96{^\circ}}}}}$
$= - \frac{16\sin{12{^\circ}}}{16\sin{12{^\circ}}}\cos{12{^\circ}\cos{24{^\circ}\cos{48{^\circ}\cos{96{^\circ}}}}}$
$= \frac{- 8\sin{24{^\circ}\cos{24{^\circ}\cos{48{^\circ}\cos{96{^\circ}}}}}}{16\sin{12{^\circ}}}$
$= \frac{- 4\sin{48{^\circ}\cos{48{^\circ}\cos{96{^\circ}}}}}{16\sin{12{^\circ}}} = - \frac{\sin{192{^\circ}}}{16\sin{12{^\circ}}}$
$= \frac{\sin{12{^\circ}}}{16\sin{12{^\circ}}} = \frac{1}{16}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.