## Task A-3.2.

Determine the minimum and maximum value of the expression

$$
\frac{1}{\sin ^{4} x+\cos ^{2} x}+\frac{1}{\sin ^{2} x+\cos ^{4} x}
$$

Determine all real numbers $x$ for which these values are achieved.
## First Solution.

Notice that

$$
\begin{aligned}
\sin ^{4} x+\cos ^{2} x & =\sin ^{2} x \cdot\left(1-\cos ^{2} x\right)+\cos ^{2} x \\
& =\sin ^{2} x+\cos ^{2} x \cdot\left(1-\sin ^{2} x\right) \\
& =\sin ^{2} x+\cos ^{4} x
\end{aligned}
$$

Therefore, we need to determine the minimum and maximum values of the expression

$$
\begin{aligned}
\frac{2}{\sin ^{4} x+\cos ^{2} x} & =\frac{2}{\sin ^{2} x \cdot\left(1-\cos ^{2} x\right)+\cos ^{2} x} \\
& =\frac{2}{1-\frac{1}{4} \sin ^{2}(2 x)}=\frac{8}{4-\sin ^{2}(2 x)}
\end{aligned}
$$

For any real number $t$, we have $-1 \leqslant \sin t \leqslant 1$, or equivalently, $0 \leqslant \sin ^{2} t \leqslant 1$.

This means that $3 \leqslant 4-\sin ^{2}(2 x) \leqslant 4$ for any real number $x$.

We conclude that the minimum value of the given expression is 2, and it is achieved if and only if $x$ is a real number such that $\sin ^{2}(2 x)=0$, i.e., if and only if $x=k \cdot \frac{\pi}{2}$, for some integer $k$.

Similarly, the maximum value of the given expression is $\frac{8}{3}$, and it is achieved if and only if $x$ is a real number such that $\sin ^{2}(2 x)=1$, i.e., if and only if $x=\frac{\pi}{4}+k \cdot \frac{\pi}{2}$, for some integer $k$.