Prove the following trigonometrical identities: \[ \cot ^{2} x \cdot \frac{\sec x-1}{1+\sin x}+\...

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Prove the following trigonometrical identities:
\[
\cot ^{2} x \cdot \frac{\sec x-1}{1+\sin x}+\sec ^{2} x \cdot \frac{\sin x-1}{1+\sec x}=0 .
\]
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