Question

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

Answer 1

Given : $\sqrt{\frac{\sec \theta -1}{\sec \theta +1}}+\sqrt{\frac{\sec \theta +1}{\sec \theta -1}}$

Question can be rewritten as ;

LHS$=\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}+\sqrt{\frac{1+\cos \theta }{1-\cos \theta }}$

Rationalize the denominators by multiplying suitable factors

$=\sqrt{\frac{1-\cos \theta }{1+\cos \theta }\times \frac{1-\cos \theta }{1-\cos \theta }}+\sqrt{\frac{1+\cos \theta }{1-\cos \theta }\times \frac{1+\cos \theta }{1+\cos \theta }}$

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

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

$=2cosec\theta$

$=RHS$

Hence proved