Question

Prove the following identify: $\frac{cosecA}{cosecA-1}+\frac{cosecA}{cosecA+a}=2{\sec }^{2}A$.

Answer 1

Given $\frac{cosecA}{cosec\theta +1}+\frac{cosecA}{cosecA-1}$

Taking LCM

$\frac{cosecA}{cosecA+1}+\frac{cosecA}{cosecA-1}=\frac{cose{c}^{2}A-cosecA+cose{c}^{2}A+cosecA}{cose{c}^{2}A-1}$

$=\frac{2cose{c}^{2}A}{cose{c}^{2}A-1}$

As $cose{c}^{2}A-1={\cot }^{2}A$

$=\frac{2cose{c}^{2}A}{{\cot }^{2}A}=\frac{\frac{2}{{\sin }^{2}A}}{\frac{{\cos }^{2}A}{{\sin }^{2}A}}=2{\sec }^{2}A$