How do you verify (sin^2x – cos^2x)/(sinx + cosx)=sinx-cosx?

Very Carefully…
The top portion of this fraction is a difference of 2 squares meaning
##sin^2(x)-cos^2(x)=(sin(x)+cos(x))(sin(x)-cos(x))## so
##(sin^2(x)-cos^2(x))/(sin(x)+cos(x))=((sin(x)+cos(x))(sin(x)-cos(x)))/(sin(x)+cos(x))##
##=sin(x)-cos(x)##