Question : Simplify $\frac{\cos ^4 \theta-\sin ^4 \theta}{\sin ^2 \theta}$.
Option 1: $1-\tan ^2 \theta$
Option 2: $\tan ^2 \theta-1$
Option 3: $\cot ^2 \theta-1$
Option 4: $1-\cot ^2 \theta$
Correct Answer: $\cot ^2 \theta-1$
Solution :
$\frac{\cos ^4 \theta-\sin ^4 \theta}{\sin ^2 \theta}$
$= \frac{(\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta)}{\sin ^2 \theta}$
$=\frac{\cos^2 \theta - \sin^2 \theta}{\sin ^2 \theta}$ [As $\cos^2 \theta + \sin^2 \theta = 1$]
$=\frac{\cos^2 \theta}{\sin ^2 \theta}-\frac{\sin^2 \theta}{\sin^2 \theta}$
$=\cot ^2 \theta-1$
Hence, the correct answer is $\cot ^2 \theta-1$.
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