(*****************************************************************)
(** * Using series definitions of cos and sin                    *)
(*****************************************************************)
Definition sin_ub (a:R) : R := sin_approx a 4.
Definition cos_lb (a:R) : R := cos_approx a 3.
Definition cos_ub (a:R) : R := cos_approx a 4.

Lemma form1 :
forall p q:R, cos p + cos q = 2 * cos ((p - q) / 2) * cos ((p + q) / 2).