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Prove that $(\text{cosec } A + \sin A)^2 + (\sec A + \cos A)^2 = 7 + \tan^2 A + \cot^2 A$.
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Solution: $LHS = \text{cosec}^2 A + \sin^2 A + 2 \text{cosec } A \sin A + \sec^2 A + \cos^2 A + 2 \cos A \sec A$
$= (\sin^2 A + \cos^2 A) + (1 + \tan^2 A) + (1 + \cot^2 A) + 4$
$= 7 + \tan^2 A + \cot^2 A = RHS$
$= (\sin^2 A + \cos^2 A) + (1 + \tan^2 A) + (1 + \cot^2 A) + 4$
$= 7 + \tan^2 A + \cot^2 A = RHS$