1. Solve the equation $\left(\sin ^{2} x+\frac{1}{\sin ^{2} x}\right)^{3}+\left(\cos ^{2} y+\frac{1}{\cos ^{2} y}\right)^{3}=16 \sin ^{2} z . \quad(5$ points)
Solution. The domain of the inequality is limited by the conditions $\sin x \neq 0, \cos y \neq 0$. Note that $u+\frac{1}{u} \geq 2$ for any positive $u$, as this inequality is equivalent to

$$
\frac{u^{2}+1}{u} \geq 2 \Leftrightarrow u^{2}+1 \geq 2 u \Leftrightarrow u^{2}+1-2 u \geq 0 \Leftrightarrow(u-1)^{2} \geq 0
$$

with equality achieved at $u=1$.

Thus, the minimum value of the left side of the equation is $2^{3}+2^{3}=16$.

The right side of the equation is no more than 16, so equality is possible only if both the left and right sides of the equation are equal to 16. This means that $\sin ^{2} x=\cos ^{2} y=\sin ^{2} z=1$, that is, $x=\frac{\pi}{2}+\pi n, y=\pi m, z=\frac{\pi}{2}+\pi k$, where $n, k, m \in Z$.

Answer. $x=\frac{\pi}{2}+\pi n, y=\pi m, z=\frac{\pi}{2}+\pi k$, where $n, k, m \in Z$.