I come across this problem in an advanced maths textbook for grade 11 in my country. And it's marked a star, which means that it's a difficult exercise, and so, no solution for this problem is given.
I can solve problems asking for which conditions do $\sin(\alpha + \beta) = \sin(\alpha) + \sin(\beta)$, and $\tan(\alpha + \beta) = \tan(\alpha) + \tan(\beta)$ hold. They are pretty easy, and straight-forward. But for this problem ($\cos(\alpha + \beta) = \cos(\alpha) + \cos(\beta)$), I have tried using all kinds of formulas, from Sum of Angles, to Sum to Product, and Double Angles, but without any luck.
So, I think there should be some glitch here that I haven't been able to spot it out.
So I hope you guys can give me some hints, or just a little push as a start.
Any help would be greatly appreciated,
Thank you very much,
And have a good day, :D