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

To find out the value of the sum, we have to use the
formula:

cos a + cos b = 2
cos[(a+b)/2]*cos[(a-b)/2]

We'll group the first and the
last term together and the middle terms together:

cos 40 +
cos 140 = 2 cos[(40+140)/2]*cos[(40-140)/2]

cos 40 + cos
140 = 2 cos[(180)/2]*cos[(-100)/2]

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

But cos 90 = 0,
so:

cos 40 + cos 140 = 0

We'll
group the middle terms together:

cos60 + cos120 = 2
cos[(60+120)/2]*cos[(60-120)/2]

cos60 + cos120 = 2 cos
90*cos (-60)

cos60 + cos120 =
0

So, the value of the sum
is:

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