Does the probability distribution of cosine transformation of uniformly distributed R.V. depend on phase shift of the cosine function?
cos(x) is periodic. Its period is 2π. Then it doesn't matter what ∆ is, the probability distribution will always be the same as the probability distribution of cos(x) over a full cycle (e.g. for x in [0, 2π]).

∆ would have mattered if x wasn't uniformly distributed, or if it wasn't distributed over a 2π range (or an integer multiple of 2π for that matter).

To me this is just obvious, I don't know if you need to prove it more rigorously though.