Question

Prove the following statements.
$\frac{cosecA}{cosecA-1}+\frac{cosecA}{cosecA+1}=2{\sec }^{3}A$.

Answer 1

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

$=cosecA\left(\frac{cosecA+1+cosecA-1}{cose{c}^2-1}\right)$

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

$=2\frac{\frac{1}{{\sin }^{2}A}}{\frac{{\cos }^{2}A}{{\sin }^{2}A}}$

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

$=RHS$

Hence proved