Question

求证sin(α+β)sin(α-β)=sin

求证sin(α+β)sin(α-β)=sin²α-sin²β

Answer 1

左边=sin(α+β)sin(α-β)
=[sinαcosβ+cosαsinβ]*[sinαcosβ-cosαsinβ]
=sin²αcos²β-cos²αsin²β
=sin²α(1-sin²β)-cos²αsin²β
=sin²α-(sin²α+cos²α)sin²β
=sin²α-sin²β
=右边