There exist real numbers $x$ and $y$ such that $\cos(x+y)=\cos x+\cos y$ I’m trying to either prove or disprove the statement that there exist real numbers x and y such that $\cos(x+y)=\cos x+\cos y$, though I quickly encountered a brick wall after expanding the LHS:
$\cos x\cos y - \sin x\sin y = \cos x + \cos y$
My question is, is there a different approach to solving this problem, or is what I started doing the right way? I couldn’t find the problem online, so I would really appreciate your responses

You can restrict yourself to the case $y=-x$.

  Then you only have to find a value $x$ such that $\cos x = 1/2$.  As $\cos 0=1$ and $\cos x=\cos (-x)$, you are done.

