Prove the given identity, where the angles involved are acute ang… - Vidyadip

Question

Prove the given identity, where the angles involved are acute angles for which the expressions are defined.
$\left( \frac { 1 + \tan ^ { 2 } A } { 1 + \cot ^ { 2 } A } \right) = \left( \frac { 1 - \tan A } { 1 - \cot A } \right) ^ { 2 }$ $= tan^2 A$

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Answer

$= \frac { 1 + \tan ^ { 2 } A } { 1 + \cot ^ { 2 } A } = \frac { 1 + \tan ^ { 2 } A } { 1 + \frac { 1 } { \tan ^ { 2 } A } } \cdot \because \cot A = \frac { 1 } { \tan A }$
$= \frac { 1 + \tan ^ { 2 } A } { \frac { \tan ^ { 2 } A + 1 } { \tan ^ { 2 } A } } = \tan ^ { 2 } A \ldots \ldots ( 1 )$
$\left( \frac { 1 - \tan A } { 1 - \cot A } \right) ^ { 2 } = \left( \frac { 1 - \tan A } { 1 - \frac { 1 } { \tan A } } \right) ^ { 2 }$
$= \left\{ \frac { 1 - \tan A } { \left( \frac { \tan A - 1 } { \tan A } \right) } \right\} ^ { 2 } = ( - \tan A ) ^ { 2 } = \tan ^ { 2 } A$ ....... (2)
(1) and (2) taken together given the result.

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