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Prove that: $\frac{1}{\cot^2 A} + \frac{1}{1 + \tan^2 A} = \frac{1}{1-\sin^2 A} - \frac{1}{\text{cosec}^2 A}$
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LHS = $\tan^2 A + \frac{1}{\sec^2 A}$
$= \tan^2 A + \cos^2 A$
RHS = $\frac{1}{\cos^2 A} - \sin^2 A$
$= \sec^2 A - \sin^2 A$
$= \tan^2 A + 1 - \sin^2 A$
$= \tan^2 A + \cos^2 A$
$\therefore$ LHS = RHS
$= \tan^2 A + \cos^2 A$
RHS = $\frac{1}{\cos^2 A} - \sin^2 A$
$= \sec^2 A - \sin^2 A$
$= \tan^2 A + 1 - \sin^2 A$
$= \tan^2 A + \cos^2 A$
$\therefore$ LHS = RHS