csc(u)/cot(u)-cot(u)/(1+csc(u)).
csc(u)/cot(u)=(1/sin(u))sin(u)/cos(u)=1/cos(u);
cot(u)/(1+csc(u))=(cos(u)/sin(u))/(1+1/sin(u))=cos(u)/(1+sin(u));
csc(u)/cot(u)-cot(u)/(1+csc(u))=1/cos(u)-cos(u)/(1+sin(u))=
(1+sin(u)-cos2(u))/(1+sin(u))=(sin(u)+sin2(u))/(1+sin(u))=sin(u).
Therefore: csc(u)/cot(u)-cot(u)/(1+csc(u))=sin(u).