Explanation

(a)
Given, $\frac{\sin{A - \sin c}}{\cos{C - \cos A}} = \cot B$
$\Rightarrow \frac{2\cos{\left( \frac{A + C}{2} \right)\sin\left( \frac{A - C}{2} \right)}}{2\sin\left( \frac{A + C}{2} \right)\sin\left( \frac{A - C}{2} \right)} = \cot B$
$\Rightarrow \cot\left( \frac{A + C}{2} \right)\cot B$
$\Rightarrow B = \frac{A + C}{2}$
Hence, $A,\ B\$and $C\$will be in AP