Explanation

(c)
$\tan{\alpha + 2\tan{2\alpha + 4\tan{4\alpha + \frac{8}{\tan{8\alpha}}}}}$
$= \tan{\alpha + 2\tan{2\alpha + 4\tan{4\alpha + \frac{4(1 - \tan^{2}{4\alpha})}{\tan{4\alpha}}}}}$
$\left\lbrack \because\ \tan{2\theta = \frac{2\tan\theta}{1 - \tan^{2}\theta}} \right\rbrack$
$= \tan{\alpha + 2\tan{2\alpha + \frac{4\tan^{2}{4\alpha + 4 - 4\tan^{2}{4\alpha}}}{\tan{4\alpha}}}}$
$= \tan{\alpha + 2\tan{2\alpha + \frac{4(1 - \tan^{2}{2\alpha})}{2\tan{2\alpha}}}}$
$= \tan{\alpha + \frac{2\tan^{2}{2\alpha + 2 - 2\tan^{2}{2\alpha}}}{\tan{2\alpha}}}$
$= \tan{\alpha + \frac{2(1 - \tan^{2}\alpha)}{2\tan\alpha}}$
$= \frac{\tan^{2}{\alpha + 1 - \tan^{2}\alpha}}{\tan\alpha} = \frac{1}{\tan\alpha} = \cot\alpha$