Question

prove that sin20 sin40 sin60 sin80=3/16

Answer 1

Answer:

sin20° sin40° sin60° sin80°=3/16

∴ LHS = sin20° sin40° sin60° sin80°

          = √3/2 (sin20° sin40° sin80°) ---------( ∵ sin60° = √3/2)

          = (√3/2 )(sin20°)(sin40° sin80°)

using the formula sin A sin B = (1/2) [ cos(A - B) - cos(A + B) ] we get,

          = (√3/2 )(sin20°) (1/2)[ cos(40°) - cos(120°) ]

          = (√3/4 )(sin20°) [ cos(40°) - cos(60°) ]

          = (√3/4 )(sin20°) [ cos(40°) - (1/2) ]

          = (√3/4 )(sin20°cos40°) - (√3/8) (sin20°)

use the formula sin A cos B = 1/2 [ sin(A + B) + sin(A - B) ]

          =  (√3/4)(1/2) [ sin60° + sin(-20°) ]+ (√3/8)sin(20°)

          = (√3/8) [ (√3 / 2) - sin(20°) ]+ (√3/8)sin(20°)

          = 3/16 - (√3/8)sin(20°) + (√3/8)sin(20°)

          = 3/16

          = RHS

∴ sin20° sin40° sin60° sin80°=3/16

Step-by-step explanation: