Prove that:$ \frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A $

To do:

We have to prove that \( \frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A \).

Solution:

We know that,

$\sec ^{2} A-\tan^2 A=1$.......(i)

$\sec A=\frac{1}{\cos A}$......(ii)

$\tan A=\frac{\sin A}{\cos A}$......(iii)

Therefore,

$\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=\frac{\sec A+1+\sec A-1}{(\sec A-1)(\sec A+1)}$

$=\frac{2 \sec A}{\sec ^{2} A-1}$

$=\frac{2 \sec A}{\tan ^{2} A}$

$=\frac{2 \times \cos ^{2} A}{\cos A \times \sin ^{2} A}$

$=\frac{2 \cos A}{\sin ^{2} A}$

$=\frac{2 \cos A}{\sin A \times \sin A}$

$=2 \cot A \operatorname{cosec} A$

$=2 \operatorname{cosec} A \cot A$

Hence proved.       

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Updated on: 10-Oct-2022

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