Question

$\frac{cosecA}{cosecA-1}+\frac{cosecA}{cosecA+1}=$

• A. ${\sec }^{2}A$
• B. $2{\sec }^{2}A$
• C. ${\cos }^{2}A$
• D. $2{\cos }^{2}A$

Answer 1

The correct option is A ${\sec }^{2}A$

Given that

$\frac{\mathrm{cosec}A}{\mathrm{cosec}A-1}+\frac{\mathrm{cosec}A}{\mathrm{cosec}A+1}$

$\Rightarrow \mathrm{cosec}A[\frac{1}{\mathrm{cosec}A-1}+\frac{1}{\mathrm{cosec}A+1}]$

$\Rightarrow \frac{1}{\sin A}[\frac{\sin A}{1-\sin A}+\frac{\sin A}{1+\sin A}]$

$\Rightarrow \frac{\sin A}{\sin A}\left[\frac{1+\sin A+1-\sin A}{(1-\sin A)(1+\sin A)}\right]$

$\Rightarrow \left[\frac{2}{1-{\sin }^{2}A}\right]\Rightarrow \frac{2}{{\cos }^{2}A}$

$\Rightarrow 2{\sec }^{2}A$

$\text{hence},\left(B\right)\text{is the correct option}$