Evaluate the definite integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{cosec} x \, dx$.

Let $I = \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{cosec} x \, dx$.
We know that $\int \operatorname{cosec} x \, dx = \log |\operatorname{cosec} x - \cot x| + C$.
Let $F(x) = \log |\operatorname{cosec} x - \cot x|$.
By the second fundamental theorem of calculus,$I = F\left(\frac{\pi}{4}\right) - F\left(\frac{\pi}{6}\right)$.
$I = \log |\operatorname{cosec} \frac{\pi}{4} - \cot \frac{\pi}{4}| - \log |\operatorname{cosec} \frac{\pi}{6} - \cot \frac{\pi}{6}|$.
Since $\operatorname{cosec} \frac{\pi}{4} = \sqrt{2}$,$\cot \frac{\pi}{4} = 1$,$\operatorname{cosec} \frac{\pi}{6} = 2$,and $\cot \frac{\pi}{6} = \sqrt{3}$,we have:
$I = \log |\sqrt{2} - 1| - \log |2 - \sqrt{3}|$.
Using the property $\log a - \log b = \log \left(\frac{a}{b}\right)$,we get:
$I = \log \left( \frac{\sqrt{2} - 1}{2 - \sqrt{3}} \right)$.