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Question
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
Solution
sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\]
\[= \frac{1}{2} \times 2\left( \sin\frac{\pi}{12} \right) \left( \sin\frac{5\pi}{12} \right)\]
\[ = \frac{1}{2}\left[ \cos\left( \frac{\pi}{12} - \frac{5\pi}{12} \right) - \cos\left( \frac{\pi}{12} + \frac{5\pi}{12} \right) \right] \left[ \because 2\sin A \sin B = \cos(A - B) - \cos(A + B) \right]\]
\[ = \frac{1}{2}\left[ \cos\left( - \frac{\pi}{3} \right) - \cos\frac{\pi}{2} \right]\]
\[ = \frac{1}{2}\left( \frac{1}{2} - 0 \right)\]
\[ = \frac{1}{4}\]
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