Prove the following identitie: cosec^4A (1 - ^4A) - 2 ^2A = 1 - Vidyadip

Question

Prove the following identitie:
$cosec^4A (1 - \cos^4A) - 2 \cot^2A = 1$

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Answer

$cosec^4A (1 - \cos^4A) - 2 \cot^2A$
$=cosec^4A (1 - \cos^2A) (1 + \cos^2A) - 2 \cot^2A$
$= cosec^4A (\sin^2A) (1 + \cos^2A) - 2 \cot^2A$
$= cosec^2A (1 + \cos^2A) - 2 \cot^2A$
$=\cos e c^2 A+\frac{\cos ^2 A}{\sin ^2 A}-2 \cot ^2 A$
$= cosec^2A + \cot^2A - 2cot^2A$
$= cosec^2A - \cot^2A$
$= 1$

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