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

Now others have given answers and hints, one technique to show there is a solution to this kind of equation is to use the intermediate value theorem - show that there are values where the LHS is less than the RHS (simply maximise RHS) and also where LHS is greater than RHS (minimise RHS). This is easy to do because the range of values on the RHS is greater than the range on the LHS and strictly contains it (use the formula for the sum of two cosines).
The fact that there are two variables means you need to take a little care to make it work, but it is a straightforward argument.
I mention this because the Intermediate Value Theorem should be in your toolkit, even if it is not always the first thing to try.
