Cosec(-x)=-cosec(x)=-1/sin(x); cot(-x)=cot(x); cos(-x)=cos(x), because these are properties of odd and even functions. So we have (1-csc(x))/(cos(x)-cot(x). Multiply top and bottom by sin(x):
(sin(x)-1)/(sin(x)cos(x)-cos(x))=
(sin(x)-1)/(cos(x)(sin(x)-1))=
1/cos(x)=sec(x) QED