Which is the value of the sum cos177+cos176+...cos4+cos3 ?

Cos177+cos176+ cos 175 + .. cos91+cos90+co89+.. ...cos4
+cos3 . To find the sum.

We know that cos (90+n) = 
-cos(90-n),

So the given series is equal to Sum
of {cos(90+n)+cos(90-n) + cos 90  for n = 1 ,2,3,....97.

=
Sum of ( 0) +cos90 , for each n = 1.2,3...87.

= 0. as cos
90 = 0.