Explanation

(b)
Given, $\cos{2x + 7 = a\ \left( 2 - \sin x \right)}$
$\Rightarrow \ \ 1 - 2\sin^{2}{x + 7 = 2a - a\sin x}$
$\Rightarrow \ 2\sin^{2}{x - a\sin{x + (2a - 8) = 0}}$
$\therefore\ \sin{x = \frac{a \pm \sqrt{( - a)^{2} - 8(2a - 8)}}{2 \times 2}}$
$= \frac{a \pm (a - 8)}{4}$
For $( + )$ sign,
$\sin{x = \frac{a - 4}{2}}$
For $( - )$ sign,
$\sin{x = 2}\$which is not possible
We know $- 1 \leq \sin{x \leq 1}$
$\therefore\ \ - 1 \leq \frac{a - 4}{2} \leq 1\ \ \ \Rightarrow \ \ 2 \leq a \leq 6$