Prove that \[ \int_{0}^{\pi / 2} \frac{\cos ^{4} x}{\left(\sin ^{4}...

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Prove that
\[
\int_{0}^{\pi / 2} \frac{\cos ^{4} x}{\left(\sin ^{4} x+\cos ^{4} x\right)} d x=\frac{\pi}{4}
\]
(IV)
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