Explanation

(b)
We have, $A = \sin^{2}{\theta + \cos^{4}\theta}$
$= \sin^{2}{\theta + \cos^{2}{\theta\cos^{2}\theta}} \leq \sin^{2}{\theta + \cos^{2}{\theta\ \ (since,\cos^{2}{\theta \leq 1)}}}$
$\Rightarrow \sin^{2}{\theta + \cos^{4}{\theta \leq 1}\ \Rightarrow A \leq 1}$
Again, $\sin^{2}{\theta + \cos^{4}{\theta = 1 - \cos^{2}{\theta + \cos^{2}{\theta + \cos^{4}\theta}}}}$
$= \cos^{4}{\theta - \cos^{2}{\theta + \cos^{4}\theta}}$
$= \cos^{4}{\theta - \cos^{2}{\theta + 1}}$
$= \left( \cos^{2}{\theta - \frac{1}{2}} \right)^{2} + \frac{3}{4} \geq \frac{3}{4}$
Hence, $\frac{3}{4} \leq A \leq 1$