Natural Log Functions Integration

Question

I've found that $\ln(|\tan(x)|)$ is the solution. I need help evaluating this problem.

Attempts:

$du = (-\csc(x)\cot(x)-\csc^2(x))dx$

Answer 1

To solve the integral of $\csc(x)$, begin by multiplying it by $\frac{\csc(x)+\cot(x)}{\csc(x)+\cot(x)}$.

Now, the integrand is $\frac{(\ (csx(x))\csc(x)+\cot(x)}{\csc(x)+\cot(x)}$

From there, substitute $u = \csc(x)+\cot(x)$, which would make $du = -\csc(x)(\csc(x)+\cot(x) dx$, making $dx=\frac{1}{-csc(x)csc(x)+cot(x)}$.

This expression cancels out with the numerator of the integrand prior to the $u-substitution$ and the denominator of the fraction is just -($\csc(x)+\cot(x)$), or $-u$. Now, the integrand is just $\frac{1}{u}$, which is equal to the $-ln|u| + C$.

Finally, undo the u-substitution to get $$-ln|\csc(x)+\cot(x)| + C$$