Explanation

(a)
$\cot^{- 1}{\left( {2.1}^{2} \right) + \cot^{- 1}{\left( {2.2}^{2} \right) + \cot^{- 1}{\left( {2.3}^{2} \right) + \ldots\infty}}}$
$\ \ \ \ \ \ \ \ \ \ \ \ = \sum_{r = 1}^{\infty}{\cot^{- 1}{(2.r^{2})}}$
$\ \ \ \ \ \ \ \ \ \ \ = \sum_{r = 1}^{\infty}{\tan^{- 1}\left( \frac{1}{2r^{2}} \right)}$
$\ \ \ \ \ \ \ \ \ \ \ = \sum_{r = 1}^{\infty}{\tan^{- 1}\left( \frac{(1 + 2r) + (1 - 2r)}{1 - (1 + 2r)(1 - 2r)} \right)}$
$\ \ \ \ \ \ \ \ \ \ = \sum_{r = 1}^{\infty}{\lbrack\tan^{- 1}{(1 + 2r) + \tan^{- 1}{(1 - 2r)}}\rbrack}$
$= \tan^{- 1}{3 - \tan^{- 1}{1 + \tan^{- 1}{5 - \tan^{- 1}{3 + \tan^{- 1}{7 - \tan^{- 1}{5 + ... + \tan^{- 1}\infty}}}}}}$
$= - \frac{\pi}{4} + \frac{\pi}{2} = \frac{\pi}{4}$