Question

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

Answer 1

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

Rationalize the denominators by multiplying suitable factors

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

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

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

$=2\sec \theta$

$=RHS$

Hence proved