(** Definition of cosinus *)
Definition cos (x:R) : R := let (a,_) := exist_cos (Rsqr x) in a.Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)).

(* Specifications: *)
Lemma simpl_sin_n : forall n:nat, sin_n (S n) / sin_n n = - / INR ((2 * S n + 1) * (2 * S n)).
Lemma Alembert_sin : Un_cv (fun n:nat => Rabs (sin_n (S n) / sin_n n)) 0.
Lemma sin_no_R0 : forall n:nat, sin_n n <> 0.