Question : The value of $\sqrt{\frac{1+\cos A}{1-\cos A}}$ is:
Option 1: $\sec A – \tan A$
Option 2: $\operatorname{cosec} A + \cot A$
Option 3: $\sec A + \tan A$
Option 4: $\operatorname{cosec} A – \cot A$
Correct Answer: $\operatorname{cosec} A + \cot A$
Solution :
Given, $\sqrt{\frac{1+\cos A}{1-\cos A}}$
$=\sqrt{\frac{1+\cos A}{1-\cos A}\times \frac{1+\cos A}{1+\cos A}}$
$=\sqrt{\frac{(1+\cos A)^2}{(1-\cos A)(1+\cos A)}}$
$=\sqrt{\frac{(1+\cos A)^2}{1-\cos^2A}}$ [Using $\small(a-b)(a+b)=a^2-b^2$]
$=\sqrt{\frac{(1+\cos A)^2}{\sin^2A}}$
$=\frac{1+\cos A}{\sin A}$
$=\frac{1}{\sin A}+\frac{\cos A}{\sin A}$
$=\operatorname{cosec}A+\cot A$
Hence, the correct answer is $\operatorname{cosec}A+\cot A$.
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