Write the value of 12 5 12 . - Vidyadip

Question

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

✓

Answer

We have,
$\sin\frac{\pi}{12}\sin\frac{5\pi}{12}=\frac{1}{2}\Big[2\sin\frac{\pi}{12}+\sin\frac{5\pi}{12}\Big]$
$=\ \frac{1}{2}\Big[\cos\Big(\frac{\pi}{12}-\frac{5\pi}{12}\Big)-\cos\Big(\frac{\pi}{12}+\frac{5\pi}{12}\Big)\Big]$
$=\ \frac{1}{2}\Big[\cos\Big(\frac{-4\pi}{12}\Big)-\cos\Big(\frac{6\pi}{12}\Big)\Big]$
$=\ \frac{1}{2}\Big[\cos\frac{\pi}{3}-\cos\frac{\pi}{2}\Big]$
$=\ \frac{1}{2}\Big[\frac{1}{2}-0\Big]$
$=\ \frac{1}{4}$
$\therefore\ \text{Value of }\sin\frac{\pi}{12}\sin\frac{5\pi}{12}=\frac{1}{4}.$

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