Proof with trivial.
intro x; unfold Omega in |- *...
case (Rle_dec (PI / 2) x); intro...
case (Rle_dec x (3 * (PI / 2))); intros...
split...
elim diff_false_true...
intro; elim diff_false_true...
Qed.

Lemma neg_cos_var : forall x : R, cos (x - PI) = (- cos x)%R.
intro; rewrite cos_minus; rewrite cos_PI; rewrite sin_PI; ring.