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Question
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
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Solution
\[ \text{ Let } f\left( x \right) = \sqrt{3} \sin x - \cos x\]
\[\text{ Dividing and multiplying by }\sqrt{3 + 1}, i . e . \text{ by 2, we get }: \]
\[ f\left( x \right) = 2\left( \frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x \right)\]
\[ \Rightarrow f(x) = 2\left( \cos\frac{\pi}{6}\sin x - \sin\frac{\pi}{6}\cos x \right)\]
\[ \Rightarrow f(x) = 2\sin\left( x - \frac{\pi}{6} \right)\]
\[\text{ Again }, \]
\[ f\left( x \right) = 2\left( \frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x \right)\]
\[ \Rightarrow f\left( x \right) = 2\left( \sin\frac{\pi}{3} \sin x - \cos\frac{\pi}{3} \cos x \right)\]
\[ \Rightarrow f\left( x \right) = - 2\cos\left( \frac{\pi}{3} + x \right)\]
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