Question

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

Answer 1

To prove:

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

Proof:

Lets take L.H.S and then equate it to R.H.S.

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

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

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

$=\frac{1+\cos A}{\sqrt{{\sin }^{2}A}}$

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

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

$=\text{cosec A}+\cot A$

$=$ RHS

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

Hence, proved.