If `y=tan^(-1) [(sqrt(1+sinx)-sqrt(1-sin x))/(sqrt(1+sin x)+sqrt(1-sin x)]]` where `0 lt x lt pi/2` find `(dy)/(dx)`

Answer

To solve the problem, we need to find the derivative of the function \( y = \tan^{-1} \left( \frac{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}} \right) \) for \( 0 < x < \frac{\pi}{2} \).

### Step 1: Simplifying the Expression
We start with the expression inside the arctangent function:

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

To simplify, we can multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{(\sqrt{1 + \sin x} - \sqrt{1 - \sin x})(\sqrt{1 + \sin x} - \sqrt{1 - \sin x})}{(\sqrt{1 + \sin x} + \sqrt{1 - \sin x})(\sqrt{1 + \sin x} - \sqrt{1 - \sin x})} \]

### Step 2: Applying the Difference of Squares
The denominator simplifies using the difference of squares:

\[ (\sqrt{1 + \sin x})^2 - (\sqrt{1 - \sin x})^2 = (1 + \sin x) - (1 - \sin x) = 2\sin x \]

The numerator expands to:

\[ (\sqrt{1 + \sin x} - \sqrt{1 - \sin x})^2 = (1 + \sin x) + (1 - \sin x) - 2\sqrt{(1 + \sin x)(1 - \sin x)} = 2 - 2\sqrt{1 - \sin^2 x} = 2 - 2\cos x \]

Thus, we can rewrite \( y \):

\[ y = \tan^{-1} \left( \frac{2(1 - \cos x)}{2\sin x} \right) = \tan^{-1} \left( \frac{1 - \cos x}{\sin x} \right) \]

### Step 3: Using Trigonometric Identities
Using the identity \( 1 - \cos x = 2\sin^2\left(\frac{x}{2}\right) \) and \( \sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) \):

\[ y = \tan^{-1} \left( \frac{2\sin^2\left(\frac{x}{2}\right)}{2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)} \right) = \tan^{-1} \left( \tan\left(\frac{x}{2}\right) \right) \]

### Step 4: Final Simplification
Since \( y = \tan^{-1}(\tan(\frac{x}{2})) \), and for \( 0 < x < \frac{\pi}{2} \), we have:

\[ y = \frac{x}{2} \]

### Step 5: Differentiating
Now, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{1}{2} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{1}{2} \]