Write the value of sin
Question:

Write the value of $\sin \frac{\pi}{12} \sin \frac{5 \pi}{12}$.

Solution:

$\sin \frac{\pi}{12} \sin \frac{5 \pi}{12}$

$=\frac{1}{2} \times 2\left(\sin \frac{\pi}{12}\right)\left(\sin \frac{5 \pi}{12}\right)$

$=\frac{1}{2}\left[\cos \left(\frac{\pi}{12}-\frac{5 \pi}{12}\right)-\cos \left(\frac{\pi}{12}+\frac{5 \pi}{12}\right)\right]$     $[\because 2 \sin A \sin B=\cos (A-B)-\cos (A+B)]$

$=\frac{1}{2}\left[\cos \left(-\frac{\pi}{3}\right)-\cos \frac{\pi}{2}\right]$

$=\frac{1}{2}\left(\frac{1}{2}-0\right)$

$=\frac{1}{4}$

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