Q. The value of the series $\cos{12{^\circ} + \cos{84{^\circ} + \cos{132{^\circ} + \cos{156{^\circ}}}}}$ is
A) $\frac{1}{2}$
B) $\frac{1}{4}$
C) $- \frac{1}{4}$
D) $- \frac{1}{2}$
Explanation
(d)
$\cos{132{^\circ} + \cos{12{^\circ} + \cos{156{^\circ} + \cos{84{^\circ}}}}}$
$= 2\cos{72{^\circ}\cos{60{^\circ} + 2\cos{120{^\circ}\cos{36{^\circ}}}}}$
$= 2\left( \frac{\sqrt{5} - 1}{4} \right)\frac{1}{2} + 2\left( \frac{- 1}{2} \right)\left( \frac{\sqrt{5} + 1}{4} \right) = \frac{- 1}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.