Cos 40 + Cos 80 + Cos 160 + cos 240 = ?...want a step by step answer.40,80,160,240 are degrees...

cos 40 + cos80 + cos 160 + cos 240
As cos ( π - x) = -cosx, and cos ( π + x) = -cosx
Hence cos(240) = cos(π + π/3 ) = -cosπ/3 = -1/2
Hence cos 40 + cos80 + cos 160 -1/2
And cos(160) = cos( π - 20) =  -cos20
Or cos 40 + cos80 + cos 160 -1/2 = cos 40 + cos80 - cos20  -1/2
Or cos(60 - 20 ) + cos(60 + 20) -cos20 -1/2
= cos60cos20 + sin60sin20 + cos60cos20 - sin60 sin20 -cos20 -1/2
= 2cos60cos20 -cos20 -1/2
= cos20 (2 cos60 -1) -1/2
As cos 60 = 1/2
Hence  cos20 (1-1) -1/2
= cos20(0) -1/2
= -1/2

Hence cos 40 + cos80 + cos 160 + cos 240 = -1/2 (Ans)