Question : Simplify the following:
$\frac{\cos x-\sqrt{3} \sin x}{2}$
Option 1: $\cos \left(\frac{\pi}{3}-x\right)$
Option 2: $\sin \left(\frac{\pi}{3}+x\right)$
Option 3: $\cos \left(\frac{\pi}{3}+x\right)$
Option 4: $\sin \left(\frac{\pi}{3}-x\right)$
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Correct Answer: $\cos \left(\frac{\pi}{3}+x\right)$
Solution :
Given, $\frac{\cos x-\sqrt{3} \sin x}{2}$
= $\frac{1}{2} \cos x-\frac{\sqrt3}{2}\sin x$
= $\cos x \cos \frac{\pi}{3} - \sin \frac{\pi}{3} \sin x$ [$\because\cos \frac{\pi}{3}=\frac{1}{2}$ and $\sin \frac{\pi}{3}=\frac{\sqrt3}{2}$]
= $\cos( \frac{\pi}{3}+x)$
Hence, the correct answer is $\cos( \frac{\pi}{3}+x)$.
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