Example 4. Let's solve the system of equations:

$$
\left\{\begin{array}{l}
\operatorname{tg}^{2} x+\operatorname{ctg}^{2} x=2 \sin ^{2} y \\
\sin ^{2} y+\cos ^{2} z=1
\end{array}\right.
$$
Solution. $\operatorname{tg}^{2} x+\operatorname{ctg}^{2} x \geqslant 2$, since $a+\frac{1}{a} \geqslant 2$ for $a>0$. Since the right-hand side of the first equation $2 \sin ^{2} y \leqslant 2$, the first equation must be satisfied for those values of $x$ and $y$ that satisfy the system:

$$
\left\{\begin{array}{l}
\operatorname{tg}^{2} x+\operatorname{ctg}^{2} x=2 \\
2 \sin ^{2} y=2
\end{array}\right.
$$

From the second equation of the original system, we obtain: $\cos ^{2} z=0$, from which $z=\frac{\pi}{2}+\pi m, m \in \mathbf{Z}$.

Solving the obtained system, we find:

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