Explanation

(a)
Let $I = \int_{}^{}{cosec\ (x - a)cosec\ x\ dx}$
$I = \int_{}^{}{\frac{\sin a}{\sin a\sin{(x - a)}\sin x}\ dx}$
$= - \frac{1}{\sin a}\int_{}^{}\left\lbrack \frac{\sin{(x - a)\cos{x - \cos{(x - a)\sin x}}}}{\sin{(x - a)\sin x}} \right\rbrack dx$
$= - \frac{1}{\sin a}\int_{}^{}\left\lbrack \cot{x - \cot\ (x - a)} \right\rbrack dx$
$= - \frac{1}{\sin a}\left\lbrack \log{|\sin{x|}} - \log\left| \sin{(x - a)} \right| \right\rbrack + c$
$= - \frac{1}{\sin a}\lbrack\log{\left| \sin x\ cosec(x - a) \right|\rbrack + c}$