sin(a)-csc(a)=?-cos(a).cot(a)

sin(a)-(1/sin(a))=-cos(a).(cos(a)/sin(a))
Multiple both side with sin(a)

sin^2(a)-1=-cos^2(a)

Multiple both side with -1

1-sin^2(a)=cos^2(a) => sin^2(a)+cos^2(a)=1

so that identity is true.I suppose you made a mistake any where.I
hope I can help you.Have a good study...

Alper Cay