Question

Prove that : $\sqrt{\frac{1-\cos A}{1+\cos A}}=\text{cosec}A-\cot A$.

Answer 1

We need to prove $\sqrt{\frac{1-\cos A}{1+\cos A}}=\text{cosec}A-\cot A$

By taking L.H.S. $=\sqrt{\frac{1-\cos A}{1+\cos A}}$

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

$=\sqrt{\frac{{(1-\cos A)}^2}{1-{\cos }^{2}A}}$

$=\sqrt{\frac{{(1-\cos A)}^2}{{\sin }^{2}A}}$

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

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

$=\text{cosec}A-\cot A=$ R.H.S.

Hence, proved