How do we know the sign for the ln?

Question

I used the $f'(x)/f(x)$ standard form to integrate the following: $$\int \frac{\cot(x)}{\ln(\sin x)} \, \mathrm{d}x = \ln|\ln\sin(x)| + C$$ However, the correct answer involved removing the modulus sign and specifying a negative sign instead: $$\int \frac{\cot(x)}{\ln(\sin x)} \, \mathrm{d}x = \ln[|\ln\sin(x)|] + C = \ln[-\ln(\sin x)] +C$$ How do we know that $\ln \sin(x)$ is negative?

Answer 1

Notice that $-1\leqslant \sin(x)\leqslant 1$ for all real number $x$. The composition $\ln(\sin(x))$ only make sense over the interval $]0,1]$ and over that interval the map $x\mapsto \ln(x)$ have range over the negative real number. However, the answer should be $\ln|\ln(\sin(x))|+C$ of course.