Let x, y, and z be real numbers such that cos(x) + cos(y) + cos(z) = sin(x) + sin(y) + sin(z) = 0.

Question

Let x, y, and z be real numbers such that cos(x) + cos(y) + cos(z) = sin(x) + sin(y) + sin(z) = 0.
Prove that cos(2x) + cos(2y) + cos(2z) = sin(2x) + sin(2y) + sin(2z) = 0.

Pip · Answer

Oh wait, I think I'm onto something and might have the solution! (Hopefully it's right... I think I'll post it on here when I'm done for confirmation)

Pip · Answer

Screenshot of my solution here (get rid of all spaces and the url should work):

lh3. googleusercontent. com/-Tpu8S83BSbc/Vy1oGcAFEiI/AAAAAAAAD-Q/LDg0yszPVOcN8pWYNyH11rLstx00HpSKQCK8B/s512/2016-05-06. png

Can someone check it please?