Question

Prove that: $\frac{\text{cosec}\theta }{\text{cosec}\theta -1}+\frac{\text{cosec}\theta }{\text{cosec}\theta +1}=2{\sec }^2\theta .$

Answer 1

$L.H.S.$

$=\frac{\cos ec\theta }{\cos ec\theta -1}+\frac{\cos ec\theta }{\cos ec\theta +1}$

$=\frac{\cos e{c}^2\theta +\cos ec\theta +\cos e{c}^2\theta -\cos ec\theta }{\cos e{c}^2\theta -1}$

$=\frac{2\cos e{c}^2\theta }{{\cot }^2\theta }$

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

$=2\frac{1}{{\cos }^2\theta }$

$=2{\sec }^2\theta R.H.S.$