Question

Prove the following:

sin 20° sin 40° sin 60° sin 80° = 3/16

Answer 1

L.H.S. = sin 20° sin 40° sin 60° sin 80° 

=  sin 20°. sin 40° .(\(\frac{\sqrt3}{2}\)).sin 80° 

=  \(\frac{\sqrt3}{2}\) . \(\frac{1}{2}\)(2 sin 40°.cos 20°) .sin 80°

=\(\frac{\sqrt3}{4}\)[cos(40°-20°)- cos(40°+20°)].sin 80°

=\(\frac{\sqrt3}{4}\)(cos 20°-cos 60°) sin 80°

=\(\frac{\sqrt3}{4}\).cos 20°.sin 80°- \(\frac{\sqrt3}{4}\)cos 60°. sin 80°