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If $y (x) = \cot^{- 1} (\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}), x \in (\frac{π}{2}, π) then \frac{dy}{dx} at x = \frac{5 π}{6} is$

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$- \frac{1}{2}$

-1

$\frac{1}{2}$

answer is A.

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Detailed Solution

$y (x) = \cot^{- 1} (\frac{\cos x / 2 + \sin x / 2 + \sin x / 2 - \cos x / 2}{\cos x / 2 + \sin x / 2 - \sin x / 2 + \cos x / 2}) = \cot^{- 1} (Tan x / 2) = \frac{π}{2} - \frac{x}{2} \Rightarrow y^{1} (x) = \frac{- 1}{2}$

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