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

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