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

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(A) Given $y = \cot^{-1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right]$.We know that $1 + \sin x

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$1/2$

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(A) Given $y = \cot^{-1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right]$.We know that $1 + \sin x = \cos^2(x/2) + \sin^2(x/2) + 2\sin(x/2)\cos(x/2) = (\cos(x/2) + \sin(x/2))^2$ and $1 - \sin x = (\cos(x/2) - \sin(x/2))^2$.Substituting these,we get $y = \cot^{-1} \left[ \frac{(\cos(x/2) + \sin(x/2)) + (\cos(x/2) - \sin(x/2))}{(\cos(x/2) + \sin(x/2)) - (\cos(x/2) - \sin(x/2))} \right]$.Simplifying the expression inside the bracket: $y = \cot^{-1} \left[ \frac{2\cos(x/2)}{2\sin(x/2)} \right] = \cot^{-1} [\cot(x/2)]$.Thus,$y = x/2$.Therefore,$\frac{dy}{dx} = \frac{1}{2}$.

If $y = \cot^{-1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right]$,then $\frac{dy}{dx} = $

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