The minimum value of $\cos^{2} (\frac{π}{3} - x) - \cos^{2} (\frac{π}{3} + x)$ is

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

$\frac{1}{2}$

$\frac{3}{2}$

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

answer is A.

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Detailed Solution

$\begin{array}{l} \cos^{2} (\frac{π}{3} - x) - \cos^{2} (\frac{π}{3} + x) \\ = \sin (\frac{π}{3} + x + \frac{π}{3} - x) . \sin (\frac{π}{3} + x - \frac{π}{3} + x) \\ = \sin \frac{2 π}{3} . \sin (2 x) \end{array} = \frac{\sqrt{3}}{2} \sin 2 x$

Minimum value

$\begin{array}{l} = \frac{\sqrt{3}}{2} (- 1) \\ = \frac{- \sqrt{3}}{2} \end{array}$

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