Explanation

(b)
Since, $a_{1},\ a_{2},\ a_{3},\ldots,a_{n}$ are in AP.
$\Rightarrow \ d = a_{2} - a_{1} = a_{3} - a_{2} = ... = a_{n} - a_{n - 1}$
$\therefore\sin{d\ (cosec\ a_{1}cosec\ a_{2} + ... + cosec\ a_{n - 1}\ cosec\ a_{n})}$
$= \ \frac{\sin{(a_{2} - a_{1})}}{\sin{a_{1}\sin a_{2}}} + ... + \frac{\sin{(a_{n} - a_{n - 1})}}{\sin a_{n - 1}\sin a_{n}}$
$= \ \frac{(\sin{a_{2}\cos{a_{1} - \cos{a_{2}\sin{a_{1})}}}}}{\sin{a_{1}\sin a_{2}}} + ... + \frac{(\sin{a_{n}\cos{a_{n - 1} + \cos{a_{n}\sin{a_{n - 1})}}}}}{\sin{a_{n - 1}\sin a_{n}}}$
$= (\cot{a_{1} - \cot{a_{2}) + (\cot{a_{2} - \cot{a_{3})}}} + ... + (\cot{a_{n - 1} - \cot{a_{n})}}}$
$= \cot{a_{1} - \cot a_{n}}$