Question

prove sin^2 20 +sin^2 40 + sin^2 80=3/2

Answer 1

sin(20)^2 + sin(40)^2 + sin(80)^2 =>
sin(20)^2 + sin(60 - 20)^2 + sin(60 + 20)^2 =>
sin(20)^2 + (sin(60)cos(20) - sin(20)cos(60))^2 + (sin(60)cos(20) + sin(20)cos(60))^2 =>
sin(20)^2 + ((1/2) * (sqrt(3) * cos(20) - sin(20)))^2 + ((1/2) * (sqrt(3) * cos(20) + sin(20)))^2 =>
sin(20)^2 + (1/4) * (3cos(20)^2 - 2sqrt(3)sin(20)cos(20) + sin(20)^2) + (1/4) * (3cos(20)^2 + 2sqrt(3)sin(20)cos(20) + sin(20)^2) =>
sin(20)^2 + (1/4) * (3cos(20)^2 - 2sqrt(3)sin(20)cos(20) + sin(20)^2 + 3cos(20)^2 + 2sqrt(3)sin(20)cos(20) + sin(20)^2) =>
sin(20)^2 + (1/4) * (6cos(20)^2 + 2sin(20)^2) =>
sin(20)^2 + (1/2) * (3cos(20)^2 + sin(20)^2) =>
(1/2) * (2sin(20)^2 + 3cos(20)^2 + sin(20)^2) =>
(1/2) * (3sin(20)^2 + 3cos(20)^2) =>
(3/2) * (sin(20)^2 + cos(20)^2) =>
(3/2) * 1 =>
3/2

Answer 2

i checked it in internet