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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Question : $\left(\frac{\tan ^3 \theta}{\sec ^2 \theta}+\frac{\cot ^3 \theta}{\operatorname{cosec}^2 \theta}+2 \sin \theta \cos \theta\right) \div\left(1+\operatorname{cosec}^2 \theta+\tan ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, is equal to:
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Question : Which of the following is equal to $[\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}]$?
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