Question

Prove the following identity-
$\frac{cosec\theta }{cosec\theta -1}+\frac{cosec\theta }{cosec\theta +1}=2{\sec }^2\theta$

Answer 1

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

Taking LCM

$\frac{cosec\theta }{cosec\theta +1}+\frac{cosec\theta }{cosec\theta -1}=\frac{cose{c}^2\theta -cosec\theta +cose{c}^2\theta +cosec\theta }{cose{c}^2\theta -1}$

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

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

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