Question : What is the value of $\sqrt{\frac{\operatorname{cosec} A+1}{\operatorname{cosec} A-1}}+\sqrt{\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}}$?
Option 1: $2 \cos A$
Option 2: $\sec A$
Option 3: $2\cos A$
Option 4: $2 \sec A$
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Correct Answer: $2 \sec A$
Solution :
$\sqrt{\frac{\operatorname{cosec} A+1}{\operatorname{cosec} A-1}}+\sqrt{\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}}$
= ${\frac{(\sqrt{\operatorname{cosec} A+1})^2+(\sqrt{\operatorname{cosec} A-1})^2}{\sqrt{(\operatorname{cosec} A-1)(\operatorname{cosec} A+1)}}}$
= ${\frac{\operatorname{cosec} A+1+{\operatorname{cosec} A-1}}{\sqrt{(\operatorname{cosec}^2A-1)}}}$
= ${\frac{2\operatorname{cosec} A}{\sqrt{\cot^2A}}}$
= ${\frac{2\operatorname{cosec} A}{\cot A}}$
= ${\frac{2\sin A}{\sin A\cos A}}$
= ${\frac{2}{\cos A}}$
= $2\sec A$
Hence, the correct answer is $2\sec A$.
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