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

To do:

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

Solution:

We know that,

$\operatorname{cosec}^2 A-\cot^2 A=1$

$\cot A=\frac{\cos A}{\sin A}$

$\operatorname{cosec} A=\frac{1}{\sin A}$

$\sec A=\frac{1}{\cos A}$

Therefore,

$\frac{\operatorname{cosec} A}{\operatorname{cosec} A-1}+\frac{\operatorname{cosec} A}{\operatorname{cosec} A+1}=\frac{\operatorname{cosec}^{2} A+\operatorname{cosec} A+\operatorname{cosec}^{2} A-\operatorname{cosec} A}{(\operatorname{cosec} A-1)(\operatorname{cosec} A+1)}$

$=\frac{2 \operatorname{cosec}^{2} A}{\operatorname{cosec}^{2} A-1}$

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

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

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

$=2 \sec ^{2} A$

Hence proved.     

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Updated on: 10-Oct-2022

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