Question

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

Answer 1

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