Question

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

Answer 1

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