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Question

Simplify: cos1+cos2+cos3+....+cos180

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Solution

cos1+cos2+cos3+...+cos177+cos178+cos179+cos180
=(cos1+cos179)+(cos2+cos178)+(cos3+cos177)+.....+cos180
We know that cos(180x)=cosx
=(cos1cos1)+(cos2cos2)+(cos3cos3)+.....+cos180
=0+0+0+....+(1)
=1.

Hence, cos1+cos2+cos3+....+cos180=1


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