proposition: Lemma cos_period : forall (x:R) (k:nat), cos (x + 2 * INR k * PI) = cos x.
proof: Proof. intros x k; induction k as [| k Hreck]. - simpl in |- *; ring_simplify (x + 2 * 0 * PI). trivial. - replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI). + rewrite cos_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *. ring_simplify; trivial. + rewrite S_INR in |- *; ring. Qed.
imports: Rbase Rfunctions SeqSeries Rtrigo_fun Rtrigo_def Rtrigo_alt Cos_rel Cos_plus ZArith_base Zcomplements Lia Lra Ranalysis1 Rsqrt_def PSeries_reg