prove identity, cotx/1+cscx=cscx-1/cotx

Question

I need to prove this identity, with full steps.

Answer 1

LHS= cot x (1 - csc x)/ (1+csc x)(1 - csc x)           (multiple both by (1 - csc x)

       = cot x (1 -csc x ) / (1 - csc^2 x )             (diff  of two sq)

       =  cot x ( 1 - csc x) / -cot^2 x                ( formula 1 + cot^2 = csc^2 x), thus 1 - csc^2 = - cot^2 x)

       = (csc x - 1)/ cot x                              (cancel the cot x and multiply -1)

       = RHS