Q. The maximum value of $\cos^{2}\left( \frac{\pi}{3} - x \right) - \cos^{2}\left( \frac{\pi}{3} + x \right)$ is
A) $- \frac{\sqrt{3}}{2}$
B) $\frac{1}{2}$
C) $\frac{\sqrt{3}}{2}$
D) $\frac{3}{2}$
Explanation
(c)
$\cos^{2}{\left( \frac{\pi}{3} - x \right) - \cos^{2}\left( \frac{\pi}{3} + x \right)}$
$= \sin\left( \frac{\pi}{3} - x + \frac{\pi}{3} + x \right)\left( \frac{\pi}{3} + x - \frac{\pi}{3} + x \right)$
$= \sin{\frac{2\pi}{3}\sin{2x = \frac{\sqrt{3}}{2}}}\sin{2x}$
[since, maximum value of $\sin{2x}$ is 1]
Its maximum value is $\frac{\sqrt{3}}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.