Prove that sin20sin40sin60sin80=3/16

We'll use the following properties of sine and cosine to prove this:







Then it's just a matter of filling it in...

sin20sin40 * sin60sin80 = 1/2(cos20 - cos60) * 1/2 (cos20 - cos140) =

1/8( cos40 + 1 - cos160 - cos120 - cos40 - cos80 + cos80 + cos200) =

1/8(1 -  cos160 - cos120 + cos200) =

1/8(1 -  cos160 - cos120 + cos160) =

1/8(1 - cos120 ) = 1/8( 1 + 1/2 ) = 3/16