Explanation

(c)
Given, $a\cos^{3}{\alpha + 3a\cos{\alpha\sin^{2}{\alpha = m}}}$
and $a\sin^{3}{\alpha + 3a\cos^{2}{\alpha\sin^{2}{\alpha = n}}}$
$\therefore(m + n) = \ a\cos^{3}{\alpha + 3a\cos{\alpha\sin^{2}{+ 3a\cos^{2}{\alpha\sin{\alpha + a\sin^{3}\alpha}}}}}$
$= a\left( \cos{\alpha + \sin\alpha} \right)^{3}$
and similarly, $(m - n) = a\left( \cos{\alpha - \sin\alpha} \right)^{3}$
$\therefore(m + n)^{2/3} + (m - n)^{2/3}$
$= a^{2/3}\ \left\{ \left( \cos{\alpha + \sin\alpha} \right)^{2} + (\cos{\alpha - {\sin{\alpha)}}^{2}} \right\}$
$= a^{2/3\ }\left\{ 2\left( \cos^{2}{\alpha + \sin^{2}\alpha} \right) \right\} = 2a^{2/3}$