Question

cot A-tan A /cot A+tan A = cos2A​

Answer 1

SOLUTION

TO PROVE

$\displaystyle \sf{ \frac{ \cot A - \tan A}{ \cot A + \tan A} = \cos 2A }$

FORMULA TO BE IMPLEMENTED

$1. \: \: \: \displaystyle \sf{ {\cos}^{2} A -{\sin}^{2} A = \cos 2 A }$

$2. \: \: \displaystyle \sf{ {\sin}^{2} A + {\cos}^{2} A = 1 }$

PROOF

$\displaystyle \sf{ \frac{ \cot A - \tan A}{ \cot A + \tan A} }$

$= \displaystyle \sf{ \frac{ \frac{ \cos A}{ \sin A} - \frac{\sin A }{\cos A} }{ \frac{ \cos A}{ \sin A} + \frac{\sin A }{\cos A}} }$

$= \displaystyle \sf{ \frac{ \frac{{\cos}^{2} A -{\sin}^{2} A }{\sin A \:\cos A } }{ \frac{{\cos}^{2} A + {\sin}^{2} A }{\sin A \:\cos A } } }$

$= \displaystyle \sf{ \frac{ {\cos}^{2} A -{\sin}^{2} A }{ {\cos}^{2} A + {\sin}^{2} A }}$

$= \displaystyle \sf{ \frac{ {\cos}2 A }{ {1 }}}$

$= \displaystyle \sf{ \cos } 2A$

Hence proved

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Answer 2

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