Example 4. Find
$$
\begin{array}{c}
\mathrm{P}_{\mathrm{n}}=\sqrt{3} \cdot \sqrt{2+\sqrt{3}} \\
\cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdots
\end{array}
$$
$$
\cdot \sqrt{2+\sqrt{2+\cdots+\sqrt{2+\sqrt{3}}}}
$$
$n$ nested radicals
Solve $\cos \frac{\pi}{6}=\frac{\sqrt{3}}{2}$,
$$
\cos \frac{\pi}{2 \cdot 6}=\sqrt{\frac{1+\cos \frac{\pi}{6}}{2}}=\sqrt{\frac{2+\sqrt{3}}{2}}
$$

In general, when $\cos \frac{\pi}{2^{\mathbf{k}-1} \cdot 6}$
$$
=\frac{1}{2} \sqrt{2+\sqrt{2+\cdots+\sqrt{2+\sqrt{3}}}} \text {, }
$$
$k$ nested radicals
$$
=\frac{1}{2} \sqrt{2+\sqrt{2+\sqrt{2+\cdots \cdots+\sqrt{2+\sqrt{3}}}}} .
$$

After calculation, we get:
$$
\begin{array}{l}
\quad P_{n}=2^{n} \cos \frac{\pi}{6} \cdot \cos \frac{\pi}{2 \cdot 6} \cdot \cos \frac{\pi}{2^{2} \cdot 6} \\
\cdots \cdot \cos \frac{\pi}{2^{n-1} \cdot 6} .
\end{array}
$$

Using the method from Example 5, we get:
$$
P_{n}=\frac{\sqrt{3}}{2 \sin \frac{\pi}{2^{n-1} \cdot 6}} .
$$