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(** * Definition of hyperbolic functions *)
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Definition sinh (x:R) : R := (exp x - exp (- x)) / 2.
Definition tanh (x:R) : R := sinh x / cosh x.

Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0.
Definition cos_in (x l:R) : Prop :=
infinite_sum (fun i:nat => cos_n i * x ^ i) l.