Prove that: (1+ ^2 A ) A cosec^2 A = A - Vidyadip

Question

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

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Answer

Using the left hand side of the equation
$\Rightarrow \frac{\left(1+\tan ^2 A\right) \cot A}{\operatorname{cosec}^2 A}$
We know $1+\tan ^2 A=\sec ^2 A$
$\frac{\left(\sec ^2 A\right) \cot A}{\operatorname{cosec}^2 A}$
$\frac{\frac{1}{\cos ^2 A}(\cot A)}{\frac{1}{\sin ^2 A}}$
$=\frac{\sin ^2 A}{\cos ^2 A}(\cot A)$
$=\tan ^2 A \times \cot A$
$=\tan ^2 A \times \frac{1}{\tan A}$
$=\tan ^2 A \times \frac{1}{\tan A}$
$=\tan A$
Hence, proved.

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