Question

cos20ᵃ+cos40°+cos140ᵃ+cos160= ?

Answer 1

$cos(20^{\circ})+cos(160^{\circ})+cos(40^{\circ})+cos(140^{\circ})=\\\\=2cos\left(\dfrac{20^{\circ}+160^{\circ}}2\right)\cdot cos\left(\dfrac{20^{\circ}-160^{\circ}}2\right)+2cos\left(\dfrac{40^{\circ}+140^{\circ}}2\right)\cdot cos\left(\dfrac{40^{\circ}-140^{\circ}}2\right)=\\\\=2cos(90^{\circ})\cdot cos(-70^{\circ})+2cos(90^{\circ})\cdot cos(-50^{\circ})=2\cdot 0\cdot cos(70^{\circ})+2\cdot 0\cdot cos(50^{\circ})=0.$

Green eyes.