3. Determine all pairs $(x, y)$ of real numbers for which

$$
\left(\sin ^{2} x+\frac{1}{\sin ^{2} x}\right)^{2}+\left(\cos ^{2} x+\frac{1}{\cos ^{2} x}\right)^{2}=12+\frac{1}{2} \sin y
$$
3. After squaring and simplifying, we get

$$
\sin ^{4} x+\cos ^{4} x+\frac{1}{\sin ^{4} x}+\frac{1}{\cos ^{4} x}=8+\frac{1}{2} \sin y
$$

5 points

This can be written in the form

$$
2\left(\sin ^{4} x+\cos ^{4} x\right)\left(1+\frac{1}{\sin ^{4} x \cos ^{4} x}\right)=16+\sin y
$$

Since $\sin ^{4} x+\cos ^{4} x=1-\frac{1}{2} \sin ^{2} 2 x$ and $\sin ^{4} x \cos ^{4} x=\frac{1}{16} \sin ^{4} 2 x$, we get

$$
\left(2-\sin ^{2} 2 x\right)\left(1+\frac{16}{\sin ^{4} 2 x}\right)=16+\sin y
$$

Since the first expression on the left side is $\geq 1$, the second is $\geq 17$, and the expression on the right side is $\leq 17$, there must be equalities everywhere.

15 points Therefore,

$$
\sin ^{2} 2 x=1, \quad \sin ^{4} 2 x=1, \quad \sin y=1
$$

so

$$
x=\frac{\pi}{4}+\frac{k \pi}{2}, \quad y=\frac{\pi}{2}+2 l \pi, \quad k, l \in \mathbf{Z}
$$

5 points