已知sin(α+β)乘以sin(α-β)=-3分之1,求sin的平方α-sin的平方β的值

cos(α+β)cos(α-β)-sin(α+β)sin(α-β)
=cos（(α+β)+(α-β)）=cos2α=cos²α-sin²α (1)式

cos(α+β)cos(α-β)+sin(α+β)sin(α-β)
=cos（(α+β)-(α-β)）=cos2β=cos²β-sin²β (2)式

(2)式-(1)式得到
2sin(α+β)sin(α-β)=cos²β-sin²β-cos²α+sin²α=2sin²α-1+1-2sin²β
=2sin²α-2sin²β
sin(α+β)sin(α-β)=sin²α-sin²β
sin(α+β)sin(α-β)=-1/3
sin²α-sin²β=-1/3

sin(α+β)×sin(α-β)=(sinαcosβ+cosαsinβ)(sinαcosβ-cosαsinβ)
=sin²αcos²β-cos²αsin²β
=sin²α(1-sin²β)-(1-sin²α)sin²β
=(sin²α-sin²αsin²β)-(sin²β-sin²αsin²β)
=sin²α-sin²αsin²β-sin²β+sin²αsin²β
=sin²α-sin²β=-1/3