Show that : cos 20 + cos 40 + ... + cos 180 = -1

We have to prove that cos 20+cos 40 + ... + cos 180 =
-1

cos 20+cos 40 + ... cos 160 + cos
180

use cos (180 - x) = -cos
x

=> cos(180 - 160) + cos (180 - 140) + ... + cos
160 + cos 180

=> -cos 160 - cos 140 - cos 120 + ...
+ cos 120 + cos 140 + cos 160 + cos 180

canceling the
negative and positive terms we are left with cos
180.

=> cos
180

=>
-1

This proves that cos 20+cos 40 + ... + cos
180 = -1