Let ,
X=sin20^circsin40^circsin60^circsin80^circX=sin20∘sin40∘sin60∘sin80∘
:.X=1/2(sin60^circsin20^circ)(2sin80^circsin40^circ)
Using ,"Product Identity"
2sinxsiny=cos(x-y)-cos(x+y),for ,x=80,y=40
:.X=1/2(sqrt3/2sin20^circ){cos(80-40)-cos(80+40)}
:.X=sqrt3/4sin20^circ{cos40^circ-cos120^circ}
:.X=sqrt3/4sin20^circ{cos40^circ+1/2}
:.X=sqrt3/8{2sin20^circcos40+sin20}
Using ,"Product Identity"
2sinxcosy=sin(x+y)+sin(x-y),for ,x=20,y=40
:.X=sqrt3/8{sin60+sin(-20)+sin20}
:.X=sqrt3/8{sqrt3/2-sin20^circ+sin20^circ}
:.X=sqrt3/8{sqrt3/2}
:.X=3/16