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If \( \alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots, \alpha_{n} \) are roots of the equation
\( x^{n}+p_{1} x^{n-1}+p_{2} x^{n-2}+\ldots+p_{n-1} x+p_{n}=0 \), then prove that
\[
\begin{aligned}
\left(1-p_{2}+\right. & \left.p_{4}+\ldots\right)^{2}+\left(p_{1}-p_{3}+p_{5} \ldots\right)^{2} \\
& =\left(1+\alpha_{1}^{2}\right)\left(1+\alpha_{2}^{2}\right)\left(1+\alpha_{3}^{2}\right) \ldots\left(1+\alpha_{n}^{2}\right) .
\end{aligned}
\]
\( \mathrm{P} \)
W
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