First slide

Compound angles

Question

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

Moderate

A

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

$\frac{1}{2}$
C

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

$\frac{3}{2}$

Solution

We have,
$\begin{array}{l} \cos^{2} (\frac{π}{3} - x) - \cos^{2} (\frac{π}{3} + x) \\ = \{\cos (\frac{π}{3} - x) + \cos (\frac{π}{3} + x)\} \{\cos (\frac{π}{3} - x) - \cos (\frac{π}{3} + x)\} \\ = 2 \cos \frac{π}{3} \cos x \times 2 \sin \frac{π}{3} \sin x \\ = \frac{\sqrt{3}}{2} \sin 2 x \leq \frac{\sqrt{3}}{2} for all x \end{array}$

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