Prove the following identity: (1+ ^2 A ) A cosec 2 = A - Vidyadip

Question

Prove the following identity:
$
\frac{\left(1+\tan ^2 A\right) \cot A}{\operatorname{cosec} 2}=\tan A
$

✓

Answer

$
\begin{aligned}
& \frac{\left(1+\tan ^2 A\right) \cot A}{\operatorname{cosec} A} \\
& =\frac{\sec ^2 A \cot A}{\operatorname{cosec}^2 A} \ldots \ldots .\left(\therefore \sec ^2 A=1+\tan ^2 A\right) \\
& =\frac{\frac{1}{\cos ^2 A} \cdot \frac{\cos A}{\sin A}}{\frac{1}{\sin ^2 A}}=\frac{1}{\frac{\cos A \sin A}{\frac{1}{\sin ^2 A}}} \\
& =\frac{\sin A}{\cos A}=\tan A
\end{aligned}
$

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