#(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#