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Question
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
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Solution
\[\sin 2x + \cos 4x\]
\[ = \sin 2x + \sin \left( \frac{\pi}{2} - 4x \right)\]
\[ = 2\sin \left( \frac{2x + \frac{\pi}{2} - 4x}{2} \right) \cos \left( \frac{2x - \frac{\pi}{2} + 4x}{2} \right) \left\{ \because \sin A + \sin B = 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\sin \left( \frac{\pi}{4} - x \right) \cos \left( 3x - \frac{\pi}{4} \right)\]
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