If A is an acute angle, then ![image.png](https://res.cloudinary.com/aptidude/image/upload/v1748021646/aptidude_questions/hityqzxfxfcelfazwr0j.png) is equal to:

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Accepted Answer

Given expression: \ \sqrt{\frac{1 - \cos A}{1 + \cos A}} + \sqrt{\frac{1 + \cos A}{1 - \cos A}} \)

Step 1: Use the identity for tangent half-angle:
$$
\tan^2 \frac{A}{2} = \frac{1 - \cos A}{1 + \cos A}
$$

Step 2: Substitute:
$$
\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \sqrt{\tan^2 \frac{A}{2}} = \tan \frac{A}{2}
$$

Similarly,
$$
\sqrt{\frac{1 + \cos A}{1 - \cos A}} = \sqrt{\cot^2 \frac{A}{2}} = \cot \frac{A}{2}
$$

Step 3: So the expression becomes:
$$
\tan \frac{A}{2} + \cot \frac{A}{2} = \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} + \frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} = \frac{\sin^2 \frac{A}{2} + \cos^2 \frac{A}{2}}{\sin \frac{A}{2} \cos \frac{A}{2}} = \frac{1}{\sin \frac{A}{2} \cos \frac{A}{2}}
$$

Step 4: Use the double angle identity:
$$
\sin A = 2 \sin \frac{A}{2} \cos \frac{A}{2} \implies \sin \frac{A}{2} \cos \frac{A}{2} = \frac{\sin A}{2}
$$

Step 5: Substitute back:
$$
\frac{1}{\sin \frac{A}{2} \cos \frac{A}{2}} = \frac{1}{\frac{\sin A}{2}} = \frac{2}{\sin A} = 2 \csc A
$$

Hence, the value of the expression is $ 2 \csc A $.

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