#cos(x+y)/sin(x-y)-2cot(x+y)#
= #cos(x+y)/sin(x-y)-(2cos(x+y))/sin(x+y)#
= #cos(x+y)[1/sin(x-y)-2/sin(x+y)]#
= #cos(x+y)[(sin(x+y)-2sin(x-y))/(sin(x-y)sin(x+y))]#
= #cos(x+y)[(sinxcosy+cosxsiny)-2(sinxcosy-cosxsiny))/((sinxcosy+cosxsiny)(sinxcosy+cosxsiny))#
= #cos(x+y){(-sinxcosy+3cosxsiny)/(sin^2xcos^2y-cos^2xsin^2y)}#
= #(cosxcosy-sinxsiny)(-sinxcosy+3cosxsiny)/(sin^2x(1-sin^2y)-(1-sin^2x)sin^2y)#
= #(cosxcosy-sinxsiny)(-sinxcosy+3cosxsiny)/(sin^2x-sin^2y)#
= #1/(sin^2x-sin^2y){(cosxcosy-sinxsiny)(-sinxcosy+3cosxsiny)}#
= #1/(sin^2x-sin^2y){-sinxcosxcos^2y+3cos^2xsinycosy+sin^2xsinycosy-3sinxcosxsin^2y}#
= #1/(sin^2x-sin^2y){-sinxcosx(cos^2y+3sin^2y)+sinycosy(3cos^2x+sin^2x)}#
= #1/(sin^2x-sin^2y){sinycosy(3-2sin^2x)-sinxcosx(3-2cos^2y)}#