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