I came across the following real analysis problem while reviewing, and I am genuinely stuck on this one:
Show that there is no non-zero polynomial $P(u,v)$ in two variables with real coefficients such that $P(x, \cos x) = 0 $ holds for all real $x$.
I just don't have any intuition about the non-existence of such a $P$. Am I missing something obvious? Starting out on the problem, why would you expect the above statement to be true?