Question:

Put $\ln |\cot x - \text{cosec} x| = t \Rightarrow \text{cosec} x \, dx = dt$<br>$\therefore \int \text{cosec} x \ln |\cot x - \text{cosec} x| dx = \int t \, dt = \frac{t^2}{2} + C = C + \frac{1}{2} \left( \ln |\text{cosec} x - \cot x| \right)^2$