Definition PI_2_aux : {z | 7/8 <= z <= 7/4 /\ -cos z = 0}.Definition PI2 := proj1_sig PI_2_aux.Definition PI := 2 * PI2.

(* Specifications: *)
Lemma cos_pi2 : cos PI2 = 0.
Lemma pi2_int : 7/8 <= PI2 <= 7/4.
Lemma cos_minus : forall x y:R, cos (x - y) = cos x * cos y + sin x * sin y.
Lemma sin2_cos2 : forall x:R, Rsqr (sin x) + Rsqr (cos x) = 1.
Lemma cos2 : forall x:R, Rsqr (cos x) = 1 - Rsqr (sin x).
Lemma sin2 : forall x:R, Rsqr (sin x) = 1 - Rsqr (cos x).
Lemma cos_PI2 : cos (PI / 2) = 0.
Lemma sin_pos_tech : forall x, 0 < x < 2 -> 0 < sin x.
Lemma sin_PI2 : sin (PI / 2) = 1.
Lemma PI_RGT_0 : PI > 0.
Lemma PI_4 : PI <= 4.
Lemma PI_neq0 : PI <> 0.
Lemma cos_PI : cos PI = -1.
Lemma sin_PI : sin PI = 0.
Lemma sin_bound : forall (a : R) (n : nat), 0 <= a -> a <= PI -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).
Lemma cos_bound : forall (a : R) (n : nat), - PI / 2 <= a -> a <= PI / 2 -> cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)).
Lemma neg_cos : forall x:R, cos (x + PI) = - cos x.
Lemma sin_cos : forall x:R, sin x = - cos (PI / 2 + x).
Lemma sin_plus : forall x y:R, sin (x + y) = sin x * cos y + cos x * sin y.
Lemma sin_minus : forall x y:R, sin (x - y) = sin x * cos y - cos x * sin y.