Доказать sin(x-y)sin(x+y)=cos^2y-cos^x

sin(x-y)sin(x+y)=cos^2y-cos2^x

Л.Ч.

sin(x-y)sin(x+y)=sin(x+(-y))sin(x+y)=

=(sinx cosy + siny cosx)*(sinx cos(-y) + sin(-y) cosx)=

=(sinx cosy + siny cosx)*(sinx cosy - siny cosx)=

=sin^2 x cos^2y - siny cosy sinx cosx - sin^2 y cos^2x + siny cosy sinx cosx=

=sin^2 x cos^2y  - sin^2 y cos^2x = ( 1- cos^2x)cos^2y -(1- cos^2y) cos^2x=

=cos^2y- cos^2y cos^2x- cos^2x+ cos^2y cos^2x=cos^2y-cos2^x

Л.Ч=П.Ч

sin(x-y)sin(x+y)=cos^2y-cos2^x

Тождество доказанно