Explanation

(b)
$\left( \frac{\cos{A + \cos B}}{\sin{A - \sin\ B}} \right)^{n} + \left( \frac{\sin{A + \sin B}}{\cos{A - \cos B}} \right)^{n}$
$\left\lbrack \frac{2\cos{\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}}{2\cos{\left( \frac{A + B}{2} \right)\sin\left( \frac{A - B}{2} \right)}} \right\rbrack^{n} + \left\lbrack \frac{2\sin{\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}}{2\sin{\left( \frac{A + B}{2} \right)\sin\left( \frac{B - A}{2} \right)}} \right\rbrack^{n}$
${= cot}^{n}{\left( \frac{A - B}{2} \right) + \cot^{n}\left( \frac{B - A}{2} \right)}$
$= \cot^{n}{\left( \frac{A - B}{2} \right) + ( - 1)^{n}\cot\left( \frac{A - B}{2} \right)}$
$= 2\cot^{n}\left( \frac{A - B}{2} \right)\ \ \ \ (\because n$ is even)