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(** * Using series definitions of cos and sin                    *)
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Definition sin_ub (a:R) : R := sin_approx a 4.
Definition cos_lb (a:R) : R := cos_approx a 3.
Definition cos_ub (a:R) : R := cos_approx a 4.

Lemma form4 :
forall p q:R, sin p - sin q = 2 * cos ((p + q) / 2) * sin ((p - q) / 2).