Trigonometric Identity?

Question

I accidentally stumbled upon an apparent trigonometric relationship.

cos(20°) = cos(40°) + cos(80°)

Please prove or disprove the equation. I’m not looking for someone to just type it into their calculator. I already know that it seems to be true but the proof is elusive.

Answer 1

Hmmmm...

cos40 +cos80 =
cos(60-20) + cos(60+20) =
cos60cos20 + sin60sin20 + cos60cos20 - sin60sin20 =
2cos60cos20 =
2*(1/2)*cos20 =
cos20

Neat!

Answer 2

cos A + cos B
= 2.cos ((A + B) / 2).cos ((A - B) / 2)
RHS
cos 40° + cos 80°
= 2.cos 60° . cos (- 20°)
= 2 x 1/2 x cos 20°
= cos 20°
= LHS, as required.