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Question
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
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Solution
\[\text{ Let } f\left( x \right) = \cos x - \sin x\]
\[\text{ Dividing and multiplying by } \sqrt{1^2 + 1^2}, i . e . \text{ by }\sqrt{2,} \text{ we get } : \]
\[ f\left( x \right) = \sqrt{2}\left( \frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x \right)\]
\[ \Rightarrow f\left( x \right) = \sqrt{2}(\cos45°\cos x - \sin45°\sin x) \]
\[ \Rightarrow f\left( x \right) = \sqrt{2}\cos\left( \frac{\pi}{4} + x \right)\]
\[\text{ Again }, \]
\[ f\left( x \right) = \sqrt{2}\left( \frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x \right)\]
\[ \Rightarrow f\left( x \right) = \sqrt{2}(\sin45°\cos x - \cos45∏\sin x)\]
\[ \Rightarrow f(x) = \sqrt{2} \sin\left( \frac{\pi}{4} - x \right)\]
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