Explanation

(c)
$\cos^{4}{x - (\lambda + 2)\cos^{2}{x - (\lambda + 3) = 0}}$
$\Rightarrow ({\cos^{2}{x)}}^{2} - (\lambda + 2)\cos^{2}{x - (\lambda + 3) = 0}$
$\therefore\cos^{2}{x = \frac{(\lambda + 2) \pm \sqrt{(\lambda + 2)^{2} + 4(\lambda + 3)}}{2}}$
$= \frac{(\lambda + 2) \pm (\lambda + 4)}{2}$
$= \lambda + 3, - 1$
$\Rightarrow \cos^{2}{x = \lambda + 3\ \ (\because\cos^{2}{x \neq - 1)}}$
But $0 \leq \cos^{2}{x \leq 1}$
$\Rightarrow 0 \leq \lambda + 3 \leq 1$
$\Rightarrow \ - 3 \leq \lambda \leq - 2$