Multiply through by 2
2sin(40) sin(80) + 2sin(80)sin(160) + 2sin (160)sin(320)
Using ....[ 2sin AsinB ] = cos(A -B) - cos(A + B)
cos ( 40 - 80) - cos(40+ 80) + cos(80 - 160) - cos(80 + 160) + cos(160 - 320) - cos(160 + 320) =
cos(-40) - cos (120) + cos (-80) - cos(240) + cos (160 - 320) - cos (480) =
[cos(480) = cos(240) = cos (120) = -1/2 ]
cos (-40) - (-1/2) + cos(-80) - (-1/2) + cos (-160) - (-1/2) =
cos (40) + cos(80) + cos (200) + 3/2 =
{Using....cos A + cos B = 2[cos ((A + B)/2)] * [ cos( (A - B)/2) ]
2[ cos ((80 + 40)/2) ] * [ cos ( ((80 - 40)/2) ] + cos(200) + 3/2 =
2cos (60) cos (20) + cos (200) + 3/2 =
2(1/2) cos (20) + cos(200) + 3/2 =
cos (20 ) + cos(200) + 3/2 =
cos(20) + cos (180 + 20) + 3/2 =
cos(20) + [ cos180cos20 - sin180sin20 ] + 3/2 =
cos(20) + (-1)cos(20) - (0)sin(20) + 3/2 =
cos (20) - cos(20) + 3/2 =
3/2
But....since we multplied the original identity by 2.....we must divide the result by 2 to get the correct answer =
3 / 4