Here's some good news to balance out the bad: If every variable that appears inside a $\sin$ or a $\cos$ is bounded, then there is something very close to a method for deciding such problems.

A basic result is that the theory of the real numbers with addition, multiplication, and the function

$S(x) = \cases{0, |x|>1\\\ \sin(x), |x|\le 1}$

is strongly model-complete. This was proved by Lou van den Dries, in a paper titled "On the Elementary Theory of Restricted Elementary Functions" (link: http://www.jstor.org/stable/2274572).

There is also an actual algorithm due to Adam Strzebonski for deciding such problems, but its correctness depends on Schanuel's conjecture, which is currently open. (link: http://dl.acm.org/citation.cfm?id=1576749.)