Integral of $\csc(x)$

Start with the identity

$$(\csc x-\cot x)(\csc x+\cot x)=\csc^2x-\cot^2x=1\;;$$

this implies that

$$|\csc x-\cot x|\cdot|\csc x+\cot x|=|\csc^2x-\cot^2x|=1\;.$$

Now use the fact that if $a,b>0$ and $ab=1$, then $\ln a+\ln b=\ln 1=0$, so $\ln a=-\ln b$ to conclude that

$$\ln|\csc x-\cot x|=-\ln|\csc x+\cot x|\;,$$

and the two answers are the same.