\(\quad \int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x d x=\)

Let \(\begin{aligned} I & =\int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x d x \\ & =\int \frac{1}{\sin (x-a) \sin x} d x \\ & =\frac{1}{\sin a} \int \frac{\sin a}{\sin (x-a) \sin x} d x \\ & =\frac{1}{\sin a} \int \frac{\sin [x-(x-a)]}{\sin (x-a) \sin x} d x \\ & =\frac{1}{\sin a} \int \frac{\sin x \cos (x-a)-\cos x \sin (x-a)}{\sin (x-a) \sin x} d x \\ & =\frac{1}{\sin a} \int[\cot (x-a)-\cot x] d x \\ & =\frac{1}{\sin a}[\log \sin (x-a)-\log (\sin x)]+c\end{aligned}\)
\(\begin{aligned} & =\frac{1}{\sin a} \log \left|\frac{\sin (x-a)}{\sin x}\right|+c \\ & =\frac{1}{\sin a} \log |\sin (x-a) \cdot \operatorname{cosec} x|+c\end{aligned}\)