Explanation

(a)
$\tan{9{^\circ} - \tan{27{^\circ} - \cot{27{^\circ} + \cot{9{^\circ}}}}}$
$= \left( \frac{\sin^{2}{9{^\circ} + \cos^{2}{9{^\circ}}}}{\cos{9{^\circ}\sin{9{^\circ}}}} \right) - \left( \frac{\sin^{2}{27{^\circ} + \cos^{2}{27{^\circ}}}}{\cos{27{^\circ}\sin{27{^\circ}}}} \right)$
$= \frac{2}{\sin{18{^\circ}}} - \frac{2}{\sin{54{^\circ}}}$
$= \frac{2}{\frac{\sqrt{5} - 1}{4}} - \frac{2}{\frac{\sqrt{5} + 1}{4}}$
$= \frac{16}{5 - 1} = 4$