Explanation

sqrt((1-sin(theta))/(1+sin(theta)))=(1-sin(theta))/cos(theta)=sec(theta)-tan(theta) (for cos(theta)>0), by multiplying top and bottom by (1-sin(theta)). The whole expression is (sec(theta)-tan(theta))-(sec(theta)-tan(theta))=0. The identity is the point; expanding is a dead end.