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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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

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