Prove: sin 20° × sin 40° × sin 60° × sin 80° = 3/16

LHS = sin 20° × sin 40° × sin 60° × sin 80°

= (√3/2) × sin 20° × sin(60° − 20°) × sin(60° + 20°)   (since sin 60° = √3/2)

= (√3/2) × sin 20° [sin²60° − sin²20°]

= (√3/2) × sin 20° [(3/4) − sin²20°]

= (√3/2) × (1/4) [3 sin 20° − 4 sin³20°]

= (√3/2) × (1/4) (sin 60°)

= (√3/2) × (1/4) × (√3/2)

= 3/16