If π/2 < x < π, then √((1 − sin x)/(1 + sin x)) + √((1 + sin x)/(1 − sin x)) is equal to(a) 2 sec x(b) −2 sec x(c) sec x(d) − sec x

Question

\[ \text{If } \frac{\pi}{2} < x < \pi,\ \text{then} \]

\[ \sqrt{\frac{1-\sin x}{1+\sin x}} + \sqrt{\frac{1+\sin x}{1-\sin x}} \]

(a) \(2\sec x\)
(b) \(-2\sec x\)
(c) \(\sec x\)
(d) \(-\sec x\)

\[ = \frac{1-\sin x}{\sqrt{1-\sin^2 x}} + \frac{1+\sin x}{\sqrt{1-\sin^2 x}} \]

\[ = \frac{1-\sin x}{|\cos x|} + \frac{1+\sin x}{|\cos x|} \]

\[ = \frac{2}{|\cos x|} \]

Since \[ \frac{\pi}{2} < x < \pi \Rightarrow \cos x < 0 \] therefore, \[ |\cos x|=-\cos x \]

\[ = \frac{2}{-\cos x} \]

\[ =-2\sec x \]

\[ \boxed{-2\sec x} \]

Correct Option: (b)

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