Question

Evaluate :
$\sqrt{\frac{1-sin\theta }{1+sin\theta }}=sec\theta -tan\theta$

Answer 1

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

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

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

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

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

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

$=\sec \theta -\tan \theta$

$=$R.H.S

Hence proved.