For convenience let s=sin(a), c=cos(a), therefore s^2+c^2=1.
So, the expression becomes: (c-(s-1))/(c+(s-1)).
Multiply top and bottom by denominator: (c^2-(s-1)^2)/(c+(s-1))^2=
(c^2-s^2+2s-1)/(c^2+s^2-2s+1+2c(s-1))=(1-2s^2+2s-1)/(2-2s+2c(s-1))=s(1-s)/((1-s)(1-c))=s/(1-c);
multiplying top and bottom by 1+c: s(1+c)/(1-c^2)=s(1+c)/s^2=(1+c)/s=1/s+c/s=cosec(a)+cot(a) QED