cot ^{ - 1}left[ frac{sqrt {1 - sin x} + sqrt | vedclass

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(D) Let $y = \cot ^{ - 1}\left[ \frac{\sqrt {1 - \sin x} + \sqrt {1 + \sin x}}{\sqrt {1 - \sin x} - \sqrt {1 + \sin x}} \right]$.Rationalizing the den

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$\pi - \frac{x}{2}$

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(D) Let $y = \cot ^{ - 1}\left[ \frac{\sqrt {1 - \sin x} + \sqrt {1 + \sin x}}{\sqrt {1 - \sin x} - \sqrt {1 + \sin x}} \right]$.Rationalizing the denominator:$y = \cot ^{ - 1}\left[ \frac{(\sqrt {1 - \sin x} + \sqrt {1 + \sin x})^2}{(\sqrt {1 - \sin x})^2 - (\sqrt {1 + \sin x})^2} \right]$Expanding the numerator and denominator:$y = \cot ^{ - 1}\left[ \frac{(1 - \sin x) + (1 + \sin x) + 2\sqrt {(1 - \sin x)(1 + \sin x)}}{(1 - \sin x) - (1 + \sin x)} \right]$$y = \cot ^{ - 1}\left[ \frac{2 + 2\sqrt {1 - \sin ^2 x}}{-2\sin x} \right] = \cot ^{ - 1}\left[ \frac{2(1 + \cos x)}{-2\sin x} \right]$Using half-angle formulas $1 + \cos x = 2\cos ^2(x/2)$ and $\sin x = 2\sin(x/2)\cos(x/2)$:$y = \cot ^{ - 1}\left[ \frac{2(2\cos ^2(x/2))}{-2(2\sin(x/2)\cos(x/2))} \right] = \cot ^{ - 1}\left[ -\frac{\cos(x/2)}{\sin(x/2)} \right]$$y = \cot ^{ - 1}\left( -\cot(x/2) \right) = \cot ^{ - 1}\left( \cot(\pi - x/2) \right) = \pi - \frac{x}{2}$.

$\cot ^{ - 1}\left[ \frac{\sqrt {1 - \sin x} + \sqrt {1 + \sin x}}{\sqrt {1 - \sin x} - \sqrt {1 + \sin x}} \right] = $

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