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Prove that: $1+\frac{1}{\tan^2 \theta} \left(1+\frac{1}{\cot^2 \theta}\right) = \frac{1}{\sin^2 \theta - \sin^4 \theta}$
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LHS = $(1 + \cot^2\theta)(1 + \tan^2\theta)$
$= \csc^2\theta \cdot \sec^2\theta$
$= \frac{1}{\sin^2\theta \cos^2\theta}$
$= \frac{1}{\sin^2\theta (1-\sin^2\theta)}$
$= \frac{1}{\sin^2\theta-\sin^4\theta} = RHS$
$= \csc^2\theta \cdot \sec^2\theta$
$= \frac{1}{\sin^2\theta \cos^2\theta}$
$= \frac{1}{\sin^2\theta (1-\sin^2\theta)}$
$= \frac{1}{\sin^2\theta-\sin^4\theta} = RHS$