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