If \( (1+x)^{n}=p_{0}+p_{1} x+p_{2} x^{2}+p_{3} x^{3}+\ldots \ldots...

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If \( (1+x)^{n}=p_{0}+p_{1} x+p_{2} x^{2}+p_{3} x^{3}+\ldots \ldots . . \), then prove that
\( \mathrm{P} \)
\[
p_{1}-p_{3}+p_{5}-\ldots \ldots=2^{n / 2} \sin \frac{n \pi}{4}
\]
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