Differentiate w.r.t. x the function `cot^(-1)""((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))),0 < x < pi/2`

Explanation

To differentiate the function \( y = \cot^{-1} \left( \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right) \) with respect to \( x \) in the interval \( 0 < x < \frac{\pi}{2} \), we will follow these steps:

### Step 1: Simplify the Argument of the Cotangent Inverse
We start with the expression inside the cotangent inverse:
\[ z = \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \]
To simplify this, we can rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{1 + \sin x} + \sqrt{1 - \sin x} \):
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