`" If "y=cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))](0ltxltpi//2)," then "(dy)/(dx)=`

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. 3
D. 1

Explanation

To solve the problem step by step, we will find the derivative of the function given by:

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

### Step 1: Simplify the expression inside the cotangent inverse function

We start by multiplying and dividing the expression by \(\sqrt{1 + \sin x} + \sqrt{1 - \sin x}\):

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