$\mathrm{K}=\sin \left(\frac{\pi}{18}\right) \sin \left(\frac{5 \pi}{18}\right) \sin \left(\frac{7 \pi}{18}\right)$
We know, $2 \sin A \sin B=\cos (A-B)-\cos (A+B)$
$K=\frac{1}{2} \cdot \sin \frac{\pi}{18}\left[2 \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18}\right]$
$=\frac{1}{2} \cdot \sin \frac{\pi}{18}\left[\cos \frac{2 \pi}{18}-\cos \frac{2}{\frac{\not / \pi} \pi}{3^{6}}\right]$
$=\frac{1}{2} \cdot \frac{1}{2}\left[2 \sin \frac{\pi}{18} \cos \frac{2 \pi}{18}-2 \sin \frac{\pi}{18} \cos \frac{2 \pi}{3}\right]$
$=\frac{1}{4}\left[\sin \left(\frac{3 \pi}{18}\right)+\sin \left(\frac{-\pi}{18}\right)-2 \sin \frac{\pi}{18} \cos \left(\pi-\frac{\pi}{3}\right)\right]$
$=\frac{1}{4}\left[\sin \frac{\pi}{6}-\sin \frac{\pi}{18}+2 \sin \frac{\pi}{18} \cdot \cos \frac{\pi}{3}\right]$
$=\frac{1}{4}\left[\sin \frac{\pi}{6}-\sin \frac{\pi}{18}+\not 2 \sin \frac{\pi}{18} \cdot \frac{1}{\not}\right]$
$=\frac{1}{4} \sin \frac{\pi}{6}=\frac{1}{4} \times \frac{1}{2}=\frac{1}{8}$