Question

The value of $16\left(sin\frac{\pi }{18}\cdot sin\frac{5\pi }{18}\cdot sin\frac{7\pi }{18}\right)$ is equal to:

Answer 1

Answer: 2

Let, $E=$ $sin\frac{\pi }{18}\cdot sin\frac{5\pi }{18}\cdot sin\frac{7\pi }{18}$
$sin\frac{7\pi }{18}=sin\left(\frac{\pi }{2}-\frac{2\pi }{18}\right)=cos\frac{2\pi }{18}$
and, $sin\frac{5\pi }{18}=sin\left(\frac{\pi }{2}-\frac{4\pi }{18}\right)=cos\frac{4\pi }{18}$
So, $E=sin\frac{\pi }{18}\cdot cos\frac{2\pi }{18}\cdot cos\frac{4\pi }{18}$
Multiply & divide by $2cos\frac{\pi }{18}$
$E=\frac{2cos\frac{\pi }{18}sin\frac{\pi }{18}\cdot cos\frac{2\pi }{18}\cdot cos\frac{4\pi }{18}}{2cos\frac{\pi }{18}}$
$E=\frac{sin\frac{8\pi }{18}}{8cos\frac{\pi }{18}}=\frac{sin\left(\frac{\pi }{2}-\frac{\pi }{18}\right)}{8cos\frac{\pi }{18}}=\frac{1}{8}$
Hence, $16E=16\left(\frac{1}{8}\right)=2$