1. (16 points) Given $a, b \neq 0$, and
$$
\begin{array}{l}
\frac{\sin ^{4} x}{a^{2}}+\frac{\cos ^{4} x}{b^{2}}=\frac{1}{a^{2}+b^{2}} . \\
\text { Find } \frac{\sin ^{100} x}{a^{100}}+\frac{\cos ^{100} x}{b^{100}} \text { . }
\end{array}
$$
$$
\begin{array}{l}
\text { 2. From } \frac{\sin ^{4} x}{a^{2}}+\frac{\cos ^{4} x}{b^{2}}=\frac{1}{a^{2}+b^{2}} \\
\begin{array}{l}
\Rightarrow\left(a^{2}+b^{2}\right)\left(\frac{\sin ^{4} x}{a^{2}}+\frac{\cos ^{4} x}{b^{2}}\right)=1 \\
=\left(\sin ^{2} x+\cos ^{2} x\right)^{2} \\
\Rightarrow \sin ^{4} x+\cos ^{4} x+\frac{b^{2}}{a^{2}} \sin ^{4} x+\frac{a^{2}}{b^{2}} \cos ^{4} x \\
=\sin ^{4} x+\cos ^{4} x+2 \sin ^{2} x \cdot \cos ^{2} x \\
\Rightarrow\left(\frac{b}{a} \sin ^{2} x-\frac{a}{b} \cos ^{2} x\right)^{2}=0 \\
\Rightarrow \frac{\sin ^{2} x}{a^{2}}=\frac{\cos ^{2} x}{b^{2}} .
\end{array}
\end{array}
$$

Let this ratio be $c$. Then from the given condition we have
$$
\begin{array}{l}
a^{2} c^{2}+b^{2} c^{2}=\frac{1}{a^{2}+b^{2}} \Rightarrow c=\frac{1}{a^{2}+b^{2}} . \\
\text { Therefore } \frac{\sin ^{100} x}{a^{100}}+\frac{\cos ^{100} x}{b^{100}}=c^{50}+c^{50}=\frac{2}{\left(a^{2}+b^{2}\right)^{50}} \text {. }
\end{array}
$$