Explanation

(c)
Since, $\tan{A + \sin{A = m}}$
and $\tan{A - \sin{A = n}}$
$\therefore m + n = 2\tan A$
and $m - n = 2\sin A$
Also, $mn = (\tan{A + \sin{A)\ (\tan{A - \sin{A) = \tan^{2}{A - \sin^{2}A}}}}}$
Now, $\frac{\left( m^{2} - \ n^{2} \right)^{2}}{mn} = \ \frac{(m + n)^{2}(m - n)^{2}}{mn}$
$= \frac{\left( 2\tan A \right)^{2}(2\sin{{A)}^{2}\ }}{\tan^{2}{A - \sin^{2}A}}$
$= \frac{16\tan^{2}{A\sin^{2}A}}{\sin^{2}{A\tan^{2}A}} = 16$