Question

Prove that $\sqrt{\frac{sec\theta -1}{sec\theta +1}}+\sqrt{\frac{sec\theta +1}{sec\theta -1}}=2cosec\theta$

Answer 1

L.H.S$=\sqrt{\frac{\sec \theta -1}{\sec \theta +1}}+\sqrt{\frac{\sec \theta +1}{\sec \theta -1}}$

$=\sqrt{\dfrac{\sec{\theta}-1}{\sec{\theta}+1}}\times\sqrt{\dfrac{\sec{\theta}-1}{\sec{\theta}-1}} +\sqrt{\dfrac{\sec{\theta}+1}{\sec{\theta}-1}}\times

\sqrt{\dfrac{\sec{\theta}+1}{\sec{\theta}+1}}$

$=\frac{\sec \theta -1}{\sqrt{{\sec }^2\theta -1}}+\frac{\sec \theta +1}{\sqrt{{\sec }^2\theta -1}}$

$=\frac{\sec \theta -1}{\sqrt{{\tan }^2\theta }}+\frac{\sec \theta +1}{\sqrt{{\tan }^2\theta }}$

$=\frac{\sec \theta -1}{\tan \theta }+\frac{\sec \theta +1}{\tan \theta }$

$=\frac{\sec \theta -1+\sec \theta +1}{\tan \theta }$

$=\frac{2\sec \theta }{\tan \theta }$

$=2\sec \theta \cot \theta$

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

$=\frac{2}{\sin \theta }$

$=2\csc \theta$