Question

Prove: $\sqrt{\frac{1-\cos A}{1+\cos A}}=cosecA-\cot A$

Answer 1

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

L.H.S Rationalizing,

$\sqrt{\frac{(1-cos\theta {)}^2}{1-co{s}^2\theta }}=\frac{1-cos\theta }{\sqrt{1-co{s}^2\theta }}=\frac{1-cos\theta }{\sqrt{si{n}^2\theta }}$

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

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

$=coses\theta -cot\theta$

$=R.H.S$

$\therefore$ $L.H.S=R.H.S$

Hence proved.