Explanation

(a)
Given, $2\sec{2\alpha = \tan{\beta + \cot\beta}}$
$\Rightarrow \ 2\sec{2\alpha = \frac{1 + \tan^{2}\beta}{\tan\beta} = \frac{\sec^{2}\beta}{\tan\beta}}$
$= \frac{2}{2\cos{\beta.\sin\beta}} = 2\ cosec\ 2\beta$
$\therefore\ \ \sec{2\alpha = \sec\left( \frac{\pi}{2} - 2\beta \right)}$
$\Rightarrow \ \ 2\alpha = 2n\pi \pm \left( \frac{\pi}{2} - 2\beta \right)$
Taking +ve sing, we have
$2(\alpha + \beta) = 2n\pi + \frac{\pi}{2}$
$\Rightarrow \ \ \alpha + \beta = n\pi + \frac{\pi}{4},\ \ n \in I$
For, $n = 0,\ \alpha + \beta = \frac{\pi}{4}$