已知sin(a+B)sin(a-B)=1/3,则sin^2a-sin^2B=?要详细过程

sin(a+b)sin(a-b)=1/3
(sinacosb+cosasinb)(sinacosb-cosasinb)=1/3
sina^2cosb^2-cosa^2sinb^2=1/3
(1-cosa^2)(1-sinb^2)-cosa^2sinb^2=1/3
[1+cosa^2sinb^2-(cosa^2+sinb^2)]-cosa^2sinb^2=1/3
1-(cosa^2+sinb^2)=1/3
sina^2-sinb^2=1/3

由积化和差公式
sin(a+b)sin(a-b)
=-[cos(a+b+a-b)-cos(a+b-a+b)]/2
=-(cos2a-cos2b)/2
=1/3
cos2a-cos2b
=-2/3
所以1-2(sina)^2-[1-2(sinb)^2]=-2/3
所以2(sinb)^2-2(sina)^2=-2/3
所以(sina)^2-(sinb)^2=1/3