Question : $\frac{\sec A(\sec A+\tan A)(1-\sin A)}{\left(\operatorname{cosec}^2 A-1\right) \sin ^2 A}$ is equal to:
Option 1: $\cot A$
Option 2: $\cos A$
Option 3: $\sec ^2 A$
Option 4: $\cos ^2 A$
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Correct Answer: $\sec ^2 A$
Solution : $\frac{\sec A(\sec A+\tan A)(1-\sin A)}{\left(\operatorname{cosec}^2 A-1\right) \sin ^2 A}$ = $\frac{\frac{1}{\cos A}(\frac{1}{\cos A}+\frac{\sin A}{\cos A})(1-\sin A)}{\left(\operatorname{cot}^2 A\right) \sin ^2 A}$ = $\frac{\frac{1}{\cos^2 A}(1+\sin A)(1-\sin A)}{\left(\frac{\cos^2A}{\sin^2A}\right) \sin ^2 A}$ = $\frac{(1-\sin^2 A)}{\cos^4 A}$ = $\frac{\cos^2 A}{\cos^4 A}$ = $\frac{1}{\cos^2A}$ = $\sec^2A$ Hence, the correct answer is $\sec^2A$.
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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:
Question : Let $0^{\circ}<\theta<90^{\circ}$, $\left(1+\cot ^2 \theta\right)\left(1+\tan ^2 \theta\right) × (\sin \theta-\operatorname{cosec} \theta)(\cos \theta-\sec \theta)$ is equal to:
Question : Which of the following is equal to $[\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}]$?
Question : $\frac{(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)}{(\sec \theta+\tan \theta)(1-\sin \theta)}$ is equal to:
Question : $\frac{1+\cos \theta-\sin ^2 \theta}{\sin \theta(1+\cos \theta)} \times \frac{\sqrt{\sec ^2 \theta+\operatorname{cosec}^2 \theta}}{\tan \theta+\cot \theta}, 0^{\circ}<\theta<90^{\circ}$, is equal to:
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