cos(2x)=cos2(x)-sin2(x)=(cos(x)-sin(x))(cos(x)+sin(x)).
tan(x)-1=sin(x)/cos(x)-1=(sin(x)-cos(x))/cos(x).
Therefore:
(tan(x)-1)/cos(2x)=(sin(x)-cos(x))/(cos(x)(cos(x)-sin(x))(cos(x)+sin(x)))=
-1/(cos(x)(cos(x)+sin(x)))=-1/(cos2(x)+sin(x)cos(x)).
Since cos(2x)=2cos2(x)-1, cos2(x)=(1+cos(2x))/2; and sin(2x)=2sin(x)cos(x), this expression can be written:
-2/(sin(2x)+cos(2x)+1).