Question

Sin20°*sin40°*sin30°*sin80°=√3/2

Answer 1

There is a mistake in the given question. There should be a factor of 8 on the LHS...

LHS = 8 * Sin20  Sin40  Sin30  Sin80
   = 8 * sin20  sin30 * sin40 * sin80
  = 4 * (Cos10 - cos50) * cos50 * sin80
  = 2 * (cos10 cos50 - cos²50) * sin80
  = 2 * [ (Cos 60+cos40) - (cos100 +1) ] * sin80
  = [ 2 cos40 - 2 cos100 - 1 ] * sin80
  =  2 cos40 sin80 - 2 cos100 sin80 - sin80 
  =  Sin120 + sin40 + 2 sin10 sin 80 - sin80
  =  √3/2 + sin40 + cos70 - cos90 - sin80 
  =  √3 /2 + sin 40 + sin20 - sin80
  =  √3 /2 + sin40 - sin80 + sin20 
  = √3 /2 + 2 sin(-20) cos60 + sin20
  = √3 /2 - sin20 + sin 20 
  = √3 /2
= RHS

Answer 2

Sin20°*sin40°*sin30°sin80°

=sin30°*sin40°*sin60°sin80°

=1/2sin*sin(60°-40°)*sin(60°+30°)

= 1/2°sin20°*(sin^2 60°-sin^2 20°)°=) sin(a-b) *sin(a+b) =sin^2a-sin^2b

=1/2*sin20°*{(√3/2)^2-sin^2 20°}

=1/2sin20°°*3/4-sin^2 20°.

=1/2(3sin20°-4sin^3 20°) /4=) 3sin-4sin^3A/4=sin3A

=1/2(sin3*20°)

=sin(3*20°)

=Sin60°
=√3/2Ans