If
K= sin (pi/18) sin (5pi/18) sin ( 7pi/18) ,
then numerical value of K is
1\frac{2\sin\left( \frac{\pi}{18}\right)\cos\left( \frac{\pi}{18}\right)\sin\left( \frac{5\pi}{18}\right)\sin\left( \frac{7\pi}{18}\right)}{2\cos\left( \frac{\pi}{18}\right)}\\ \\ \\ \frac{2\sin\left( \frac{2\pi}{18}\right)\sin\left( \frac{5\pi}{18}\right)\cos\left( \frac{2\pi}{18}\right)}{4\cos\left( \frac{\pi}{18}\right)} \\ \\ \\ \frac{2\sin\left( \frac{4\pi}{18}\right)\sin\left( \frac{5\pi}{18}\right)}{8\cos\left( \frac{\pi}{18}\right)} \\ \\ \\ \frac{2\sin\left( \frac{4\pi}{18}\right)\cos\left( \frac{4\pi}{18}\right)}{8\cos\left( \frac{\pi}{18}\right)} \\ \\ \\ \frac{\cos\left( \frac{\pi}{18}\right)}{8\cos\left( \frac{\pi}{18}\right)} \\ \\ \\ \boxed{Ans.=\frac{1}{8}}
1ya i got it. just seconds after posting it,i again thought and got it easily.
instead of doing it, we have
K=sin10° sin 50° sin 70°
= cos80°cos 40° cos 20°
= ( (2sin 20°cos 20°) cos 40° cos80° )/ ( 2sin 20°)
after that the same procedure you applied,and we get (1/8) finally.