Example 4 Simplify $\frac{\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}$. The result is ( ). [4]
(A) $\sqrt{2}$
(B) $\frac{\sqrt{2}}{2}$
(C) 2
(D) $\frac{1}{2}$
(2010, National Junior High School Mathematics League, Jiangxi Province Preliminary)
Notice
$$
\begin{array}{l}
13+\sqrt{48}=(2 \sqrt{3})^{2}+1+4 \sqrt{3}=(1+2 \sqrt{3})^{2}, \\
5-(1+2 \sqrt{3})=4-2 \sqrt{3}=(\sqrt{3}-1)^{2}, \\
3+(\sqrt{3}-1)=2+\sqrt{3}=\frac{4+2 \sqrt{3}}{2} \\
=\frac{(\sqrt{3}+1)^{2}}{2}=\left(\frac{\sqrt{6}+\sqrt{2}}{2}\right)^{2} .
\end{array}
$$

Therefore, the original expression $=\frac{1}{2}$.
Hence, the correct choice is (D).