Cos(x-y)(tgx tgy-1)+(1+tgx tgy)cos(x+y)
30 баллов!

$cos(x-y)*(tgx*tgy-1)+(1+tgx*tgy)*cos(x+y)=$

$=cos(x-y)*(frac{sinx}{cosx}*frac{siny}{cosy}-1)+(1+frac{sinx}{cosx}*frac{siny}{cosy})*cos(x+y)=$

$=cos(x-y)*frac{sinx*siny-cosx*cosy}{cosx*cosxy}+cos(x+y)*frac{cosx*cosy+sinx*siny}{cosx*cosy}=$

$=frac{cos(x-y)*(sinx*siny-cosx*cosy)}{cosx*cosy}+frac{cos(x+y)*(cosx*cosy+sinx*siny)}{cosx*cosy}=$

$=frac{cos(x-y)*(sinx*siny-cosx*cosy)+cos(x+y)*(cosx*cosy+sinx*siny)}{cosx*cosy}=$

$=cos(x-y)(sinxsiny-cosxcosy)+cos(x+y)(cosxcosy+sinxsiny)$

$=(cos(x)cos(y)+sin(x)sin(y))(sin(x)sin(y)-cos(x)cos(y))+$
$+(cos(x)cos(y)-sin(x)sin(y))*(cos(x)cos(y)+sin(x)sin(y))=$

$=sin^2(x)sin^2(y)-cos^2(x)cos^2(y)+cos^2(x)cos^2(y)-sin^2(x)sin^2(y)=$
$=0$