Question

cos20+cos100+cos140=cos52+cos68+cos172

Answer 1

cos(α±β)=cosαcosβ∓sinαsinβcos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β

and

cos(180∘−α)=−cosα.cos⁡(180∘−α)=−cos⁡α.

Hence

cos52∘+cos68∘+cos172∘



=cos(60∘−8∘)+cos(60∘+8∘)+cos(180∘−8∘)=2cos60∘cos8∘+(−cos8∘)=2⋅12⋅cos8∘−cos8
=0