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 $2\sec A$.