Question : What is the value of $\frac{\sin \theta+\cos \theta}{\sin \theta-\cos \theta}+\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}$?
Option 1: $\frac{1}{\left(\sin ^2 \theta-\cos ^2 \theta\right)}$
Option 2: $2\left(\sin ^2 \theta-\cos ^2 \theta\right)$
Option 3: $\frac{2}{\left(\sin ^2 \theta-\cos ^2 \theta\right)}$
Option 4: $\sin ^2 \theta-\cos ^2 \theta$
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Correct Answer: $\frac{2}{\left(\sin ^2 \theta-\cos ^2 \theta\right)}$
Solution :
$\frac{\sin \theta+\cos \theta}{\sin \theta-\cos \theta}+\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}$
Taking the LCM of the denominators:
= $\frac{(\sin \theta+\cos \theta)^2 + (\sin \theta-\cos \theta)^2}{\sin^2 \theta-\cos^2 \theta}$
= $\frac{(\sin^2 \theta+\cos^2 \theta+2\sin \theta \cos \theta) + (\sin^2 \theta+\cos^2 \theta-2\sin \theta \cos \theta)}{\sin^2 \theta-\cos^2 \theta}$
= $\frac{(1+2\sin \theta \cos \theta) + (1-2\sin \theta \cos \theta)}{\sin^2 \theta-\cos^2 \theta}$
= $\frac{2}{\sin^2 \theta-\cos^2 \theta}$
Hence, the correct answer is $\frac{2}{\sin^2 \theta-\cos^2 \theta}$.
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