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