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

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 $2cos\frac{\pi }{18}$
$E=\frac{2 cos \frac{\pi }{18} sin \frac{\pi }{18} \cdot cos \frac{2 \pi }{18} \cdot cos \frac{4 \pi }{18}}{2 cos \frac{\pi }{18}}$
$E=\frac{sin \frac{8 \pi }{18}}{8 cos \frac{\pi }{18}}=\frac{sin \left(\frac{\pi }{2} - \frac{\pi }{18}\right)}{8 cos \frac{\pi }{18}}=\frac{1}{8}$
Hence, $16E=16\left(\frac{1}{8}\right)=2$