\(\int \frac{1}{\sin (x-a) \sin x} \mathrm{~d} x=\)

Let \(\mathrm{I}=\int \frac{1}{\sin (x-\mathrm{a}) \sin x} \mathrm{~d} x\)
Put \(x-\mathrm{a}=\mathrm{t} \rightarrow x=\mathrm{a}+\mathrm{t}\) \(\mathrm{d} x=\mathrm{dt}\)
\(\therefore I =\int \frac{1}{\sin t \cdot \sin (a+t)} d t \)
\( =\frac{1}{\sin a} \int \frac{\sin a}{\sin t \cdot \sin (a+t)} d t \)
\( =\frac{1}{\sin a} \int \frac{\sin ((a+t)-t)}{\sin (a+t) \cdot \sin t} d t \)
\( =\frac{1}{\sin a}\left[\int \frac{\sin (a+t) \cos t}{\sin (a+t) \sin t} d t-\int \frac{\sin t \cos (a+t)}{\sin (a+t) \sin t} d t\right] \)
\( =\frac{1}{\sin a}\left[\int \cot t d t-\int \cot (a+t) d t\right] \)
\( =\operatorname{cosec} a[\log |\sin t|-\log |\sin (a+t)|]+c \)
\( =\operatorname{cosec} a\left[\log \left|\frac{\sin t}{\sin (a+t)}\right|\right]+c \)
\( =\operatorname{cosec} a[\log (\sin (x-a) \cdot \operatorname{cosec} x)]+c\)