Explanation

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