Question

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. $\frac{3}{2}$

Answer 1

Answer: (C)

$\frac{\sqrt{3}}{2}$

Answer 2

$\cos^{2}\left( \frac{\pi}{3} - x \right) - \cos^{2}\left( \frac{\pi}{3} + x \right)$

$= \left\{ \cos\left( \frac{\pi}{3} - x \right) + \cos\left( \frac{\pi}{3} + x \right) \right\}\left\{ \cos\left( \frac{\pi}{3} - x \right) - \cos\left( \frac{\pi}{3} + x \right) \right\}$

$= \left\{ 2\cos\frac{\pi}{3}\cos x \right\}\left\{ 2\sin\frac{\pi}{3}\sin x \right\} = \sin\frac{2\pi}{3}\sin 2x = \frac{\sqrt{3}}{2}\sin 2x$

Its maximum value is $\frac{\sqrt{3}}{2},\{ - 1 \leq \sin 2x \leq 1\}$.