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