Example 8. Given $1 x=(3+2 \sqrt{2})^{-1}$, $y=(3-2 \sqrt{2})^{-1}$, find the value of $(x+1)^{-1}+(y+1)^{-1}$.
$$
\begin{array}{c}
\text { Sol } \because x=(3+2 \sqrt{2})^{-1} \\
=\frac{3-2 \sqrt{2}}{(3+2 \sqrt{2})(3-2 \sqrt{2})}=3-2 \sqrt{2},
\end{array}
$$
$\therefore x, y$ are conjugate radicals, thus we have
$$
\begin{array}{l}
y=x^{-1}=3+2 \sqrt{2} . \\
\therefore(x+1)^{-1}+(y+1)^{-1}=(4-2 \sqrt{2})^{-1} \\
\quad+(4+2 \sqrt{2})^{-1}=1 .
\end{array}
$$