(pi/9)=20°, (2pi)/9=40°, (4pi)/9=80°
LHS=sin(pi/9)*sin((2pi)/9)*sin((4pi)/9)
=sin20°sin40°sin80°
=(2sin20°sin40°sin80°)/2
=((cos(40°-20°)-cos(40°+20°))*sin80°)/2
=((cos20°-cos60°)*sin80°)/2
=((cos20°-1/2)*sin80°)/2
=(((2cos20°-1)/2)*sin80°)/2
=((2cos20°-1)*sin80)/4
=(2cos20°sin80°-sin80°)/4
=(sin(80°+20°)+sin(80°-20°)-sin80°)/4
=(sin100°+sin60°-sin80°)/4
=(sin(180°-100°)+sin60°-sin80°)/4
=(cancel(sin80°)+sin60°cancel(-sin80°))/4
=(sqrt(3)/2)/4=sqrt(3)/8=RHS