# Library Coq.Reals.Rtrigo_reg

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo.
Require Import Ranalysis1.
Require Import PSeries_reg.
Open Local Scope nat_scope.
Open Local Scope R_scope.
Lemma CVN_R_cos :
forall fn:nat -> R -> R,
fn = ( fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N ) ) ->
CVN_R fn.
Lemma continuity_cos : continuity cos.
Lemma continuity_sin : continuity sin.
Lemma CVN_R_sin :
forall fn:nat -> R -> R,
fn =
( fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N ) ) ->
CVN_R fn.
(sin h)/h -> 1 when h -> 0
Lemma derivable_pt_lim_sin_0 : derivable_pt_lim sin 0 1.
((cos h)-1)/h -> 0 when h -> 0
Lemma derivable_pt_lim_cos_0 : derivable_pt_lim cos 0 0.
Theorem derivable_pt_lim_sin : forall x:R, derivable_pt_lim sin x (cos x).
Lemma derivable_pt_lim_cos : forall x:R, derivable_pt_lim cos x (- sin x).
Lemma derivable_pt_sin : forall x:R, derivable_pt sin x.
Lemma derivable_pt_cos : forall x:R, derivable_pt cos x.
Lemma derivable_sin : derivable sin.
Lemma derivable_cos : derivable cos.
Lemma derive_pt_sin :
forall x:R, derive_pt sin x (derivable_pt_sin _) = cos x.
Lemma derive_pt_cos :
forall x:R, derive_pt cos x (derivable_pt_cos _) = - sin x.