Calculate the value of the sum cos40 + cos60 + cos120 + cos140.

cos40 + cos60 +
cos120 + cos140

Let us rewrite:

cos40 + cos60 +
cos(180-60) + cos(140)

But cos(180-60) = - cos60

==> cos40 + cos60 - cos60 + cos(140)

==> cos40 +
cos(140)

We know that:

cosa + cosb = 2cos(a+b)/2*
cos(a-b)/2

cos 40 + cos(140) = 2cos(90)*cos(-50) = 0

==> cos40 + cos60 + cos120 + cos140 = 0