Explanation

(a)
$\sum_{m = 1}^{n}{\tan^{- 1}\left( \frac{2m}{m^{4} + m^{2} + 2} \right)}$
$\ \ \ \ \ = \sum_{m = 1}^{n}{\tan^{- 1}\left( \frac{\left( m^{2} + m + 1 \right) - (m^{2} - m + 1)}{1 + \left( m^{2} + m + 1 \right)(m^{2} - m + 1)} \right)}$
$\ \ \ \ = \sum_{m = 1}^{n}{\lbrack\tan^{- 1}{\left( m^{2} + m + 1 \right) - \tan^{- 1}{(m^{2} - m + 1)}}\rbrack}$
$\ \ \ = \tan^{- 1}{\left( n^{2} + n + 1 \right) - \tan^{- 1}1}$
$\ \ \ = \tan^{- 1}\left( \frac{n^{2} + n}{2 + n^{2} + n} \right)$