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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Question : The value of $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}$ is:
Question : $\frac{(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)}{(\sec \theta+\tan \theta)(1-\sin \theta)}$ is equal to:
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:
Question : If $\frac{\sec \theta-\tan \theta}{\sec \theta+\tan \theta}=\frac{1}{7}, \theta$ lies in first quadrant, then the value of $\frac{\operatorname{cosec} \theta+\cot ^2 \theta}{\operatorname{cosec} \theta-\cot ^2 \theta}$ is:
Question : Which of the following is equal to $\frac{1}{\tan \theta}+\tan \theta$?
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