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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