Question : $ \sqrt{\frac{1+\cos \theta}{1-\cos \theta}}+\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=$________.
Option 1: $2 \sin \theta$
Option 2: $2 \cos \theta$
Option 3: $ 2 \operatorname{cosec} \theta $
Option 4: $2 \sec \theta$
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Correct Answer: $ 2 \operatorname{cosec} \theta $
Solution :
Given: $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}+\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}$
= $ \frac{(1+\cos \theta) + (1-\cos \theta)}{\sqrt{(1-\cos \theta)(1+\cos \theta)}}$
= $ \frac{2}{\sqrt{(1-\cos^2 \theta)}}$
= $\frac{2}{\sin \theta}$
= $ 2 \operatorname{cosec} \theta $
Hence, the correct answer is $ 2 \operatorname{cosec} \theta $.
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