Explanation

(b)
We have, $\sum_{}^{}{x_{1} = \sin{2\beta,\ \sum_{}^{}{x_{1}x_{2} = \cos{2\beta}}}}$
$\sum_{}^{}{x_{1}x_{2}x_{3} = \cos\beta}$ and $x_{1}x_{2}x_{3}x_{4} = - \sin\beta$
$\tan^{- 1}{x_{1} + \tan^{- 1}{x_{2} + \tan^{- 1}{x_{3} + \tan^{- 1}x_{4}}}}$
$= \tan^{- 1}\left( \frac{\sum_{}^{}{x_{1} - \sum_{}^{}{x_{1}x_{2}x_{3}}}}{1 - \sum_{}^{}{x_{1}x_{2} + x_{1}x_{2}x_{3}x_{4}}} \right)$
$= \tan^{- 1}\left( \frac{\sin{2\beta - \cos\beta}}{1 - \cos{2\beta - \sin\beta}} \right)$
$= \tan^{- 1}\left( \frac{(2\sin{\beta - 1)\cos\beta}}{\sin{\beta(2\sin{\beta - 1})}} \right) = \tan^{- 1}{(\cot\beta)}$
$= \tan^{- 1}\left\lbrack \tan\left( \frac{\pi}{2} - \beta \right) = \frac{\pi}{2} - \beta \right\rbrack$