Question

已知cos²α-cos²β=a,那么sin（α+β）×sin（α-β）等于？

Answer 1

sin（α+β）×sin（α-β）由积化和差公式得
sin（α+β）×sin（α-β）=(cos2β-cos2α)/2
使用倍角公式得
(cos2β-cos2α)/2=(2cos^2β-1-2cos^2α+1)/2=cos²β-cos²α=-a

Answer 2

sin(α+β)sin(α-β)
=(sinacosβ+cosasinβ)(sinacosβ-cosasinβ)
=(sinacosβ)^2-(cosasinβ)^2
=sin^2a(1-sin^2β)-(1-sin^2a)sin^2β
=sin^2a-sin^2β
=-[cos²α-cos²β]
=-a