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Question
Prove that:
Solution
Consider LHS:
\[\cos \left( \frac{3\pi}{4} + x \right) - \cos \left( \frac{3\pi}{4} - x \right)\]
\[ = - 2\sin\left\{ \frac{\left( \frac{3\pi}{4} + x \right) + \left( \frac{3\pi}{4} - x \right)}{2} \right\} \sin \left\{ \frac{\left( \frac{3\pi}{4} + x \right) - \left( \frac{3\pi}{4} - x \right)}{2} \right\} \left\{ \because \cos A - \cos B = - 2\sin \left( \frac{A + B}{2} \right) \sin\left( \frac{A - B}{2} \right) \right\}\]
\[ = - 2\sin\frac{3\pi}{4} \sin x\]
\[ = - 2\sin \left( \pi - \frac{\pi}{4} \right) \sin x\]
\[ = - 2\sin \frac{\pi}{4} \sin x\]
\[ = - \sqrt{2}\sin x\]
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