Algebraically, how are $-\mathrm{ln}|\mathrm{csc}x+\mathrm{cot}x|+C$ and $\mathrm{ln}|\mathrm{csc}x-\mathrm{cot}x|+C$ equal?

I know both of these are the answer to $\int \mathrm{csc}x\text{}dx$, and I am able to work them out with calculus using the formulas:

$\int \mathrm{csc}x\text{}dx$

$=\int \mathrm{csc}x\frac{\mathrm{csc}x-\mathrm{cot}x}{\mathrm{csc}x-\mathrm{cot}x}\text{}dx$

and:

$=\int \mathrm{csc}x\frac{\mathrm{csc}x+\mathrm{cot}x}{\mathrm{csc}x+\mathrm{cot}x}\text{}dx$

Still, when looking at the results, $-\mathrm{ln}|\mathrm{csc}x+\mathrm{cot}x|+C$ and $\mathrm{ln}|\mathrm{csc}x-\mathrm{cot}x|+C$ , I don't see how these are algebraically equivalent. Perhaps I'm just unaware of some algebra rule (that is likely!). I tried using the Laws of Logs and that doesn't help. Or maybe I'm missing some trig trick.

I know both of these are the answer to $\int \mathrm{csc}x\text{}dx$, and I am able to work them out with calculus using the formulas:

$\int \mathrm{csc}x\text{}dx$

$=\int \mathrm{csc}x\frac{\mathrm{csc}x-\mathrm{cot}x}{\mathrm{csc}x-\mathrm{cot}x}\text{}dx$

and:

$=\int \mathrm{csc}x\frac{\mathrm{csc}x+\mathrm{cot}x}{\mathrm{csc}x+\mathrm{cot}x}\text{}dx$

Still, when looking at the results, $-\mathrm{ln}|\mathrm{csc}x+\mathrm{cot}x|+C$ and $\mathrm{ln}|\mathrm{csc}x-\mathrm{cot}x|+C$ , I don't see how these are algebraically equivalent. Perhaps I'm just unaware of some algebra rule (that is likely!). I tried using the Laws of Logs and that doesn't help. Or maybe I'm missing some trig trick.