Question : The given expression is equal to:
$1-\frac{\tan ^2 \phi}{\sec ^2 \phi}$
Option 1: $\operatorname{sin}^2 \phi\cos ^2 \phi$
Option 2: $\operatorname{sin}^2 \phi\cot ^2 \phi$
Option 3: $\cot^2 \phi \cos^2 \phi$
Option 4: $\tan^2 \phi\cos^2 \phi$
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Correct Answer: $\operatorname{sin}^2 \phi\cot ^2 \phi$
Solution :
We know, $\sec^2 \phi-\tan^2 \phi$ = 1 and $\frac{\cos^2 \phi}{\sin^2 \phi}$ = $\cot^2 \phi$
So, $1-\frac{\tan^2 \phi}{\sec^2 \phi} $
= $\frac{\sec^2 \phi-\tan^2 \phi}{\sec^2 \phi}$
= $\frac{1}{\sec^2 \emptyset}$
= $\cos^2 \phi$
= $\frac{\sin^2 \phi\times{\cos^2 \phi}}{\sin^2 \phi}$
= $\sin^2 \phi\cot^2 \phi$
Hence, the correct answer is $\sin^2 \phi\cot^2 \phi$.
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