Explanation

(b)
$\cot^{- 1}{9 + cosec^{- 1}\frac{\sqrt{41}}{4} = \tan^{- 1}{\frac{1}{9} + \tan^{- 1}\frac{1}{\sqrt{\frac{41}{16} - 1}}}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\lbrack \because cosec^{- 1}x = \tan^{- 1}\frac{1}{\sqrt{x^{2} - 1}} \right\rbrack$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \tan^{- 1}{\frac{1}{9} + \tan^{- 1}\frac{4}{5}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \tan^{- 1}\left( \frac{\frac{1}{9} + \frac{4}{5}}{1 - \frac{1}{9}.\frac{4}{5}} \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \tan^{- 1}{\left( \frac{41}{41} \right) = \frac{\pi}{4}}$