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

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)).