Question

Prove the following:

sin 20° sin 40° sin 80° = √3/8

Answer 1

L.H.S. = sin 20°. sin 40°. sin 80° 

= sin 20°. sin 40°. sin 80°

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

= \(\frac{1}{2}\)[cos(40° – 20°) – cos (40° + 20°)] . sin 80°

= \(\frac{1}{2}\)(cos 20° – cos 60°) sin 80°

= \(\frac{1}{2}\). cos 20° . sin 80° – \(\frac{1}{2}\). cos 60° . sin 80°

= \(\frac{1}{2\times2}\)(2 sin 80° . cos 20°) -\(\frac{1}{2\times2}\).sin 80°

= \(\frac{1}{2}\)[sin(80° + 20°) + sin (80° – 20°)] –\(\frac{1}{2}\). sin 80°