1. Proof

Strategy: The strategy is not obvious.  Multiply and divide by (csc x + cot x); use Substitution.
 

csc x dx =  csc x

csc x + cot x  csc x + cot x

dx

set
  u = csc x + cot x
then we find
  du = (- csc x cot x - csc² x) dx

substitute du = (- csc x cot x - csc² x) dx, u = csc x + cot x
 

csc x

csc x + cot x  csc x + cot x

dx = -

(- csc2 x - csc x cot x) dx  csc x + cot x

 

= -

du u

solve integral

= - ln |u| + C

substitute back u=csc x + cot x

= - ln |csc x + cot x| + C
Q.E.D.

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