If `y="tan"^(-1)((sqrt(1+sinx)+sqrt(1-sinx)))/((sqrt(1+sinx)-sqrt(1-sinx)))," find "(dy)/(dx).`

Options

A. `{:{((1)/(2)", " "cos"(x)/(2)gt"sin"(x)/(2)),((-(1)/(2)",","cos"(x)/(2)lt"sin"(x)/(2))),("doesnotexist," x={npi},n in "integer"):}`
B. `{:{(-(1)/(2)", " "cos"(x)/(2)gt"sin"(x)/(2)),(((1)/(2)",","cos"(x)/(2)lt"sin"(x)/(2))),("doesnotexist," x={npi},n in "integer"):}`
C. `{:{(-(1)/(2)", " "cos"(x)/(2)ge"sin"(x)/(2)),(((1)/(2)",","cos"(x)/(2)lt"sin"(x)/(2))):}`
D. None of these

Answer

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

### Step-by-Step Solution:

1. **Let \( z \) be the expression inside the arctangent:**
\[ z = \frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}} \]

2. **Rationalize the expression for \( z \):**
Multiply the numerator and the denominator by \( \sqrt{1+\sin x} - \sqrt{1-\sin x} \):
\[ z = \frac{(\sqrt{1+\sin x} + \sqrt{1-\sin x})(\sqrt{1+\sin x} - \sqrt{1-\sin x})}{(\sqrt{1+\sin x} - \sqrt{1-\sin x})(\sqrt{1+\sin x} - \sqrt{1-\sin x})} \]

3. **Simplify the numerator using the difference of squares:**
\[ \text{Numerator: } (1+\sin x) - (1-\sin x) = 2\sin x \]
\[ \text{Denominator: } (\sqrt{1+\sin x} - \sqrt{1-\sin x})^2 = (1+\sin x) + (1-\sin x) - 2\sqrt{(1+\sin x)(1-\sin x)} = 2 - 2\sqrt{1-\sin^2 x} = 2 - 2\cos x \]

4. **Combine the results:**
\[ z = \frac{2\sin x}{2(1 - \cos x)} = \frac{\sin x}{1 - \cos x} \]

5. **Recognize the identity:**
Using the identity \( 1 - \cos x = 2\sin^2\left(\frac{x}{2}\right) \):
\[ z = \frac{\sin x}{2\sin^2\left(\frac{x}{2}\right)} = \frac{2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)}{2\sin^2\left(\frac{x}{2}\right)} = \frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} = \cot\left(\frac{x}{2}\right) \]

6. **Substituting back into \( y \):**
\[ y = \tan^{-1}(\cot\left(\frac{x}{2}\right)) = \frac{\pi}{2} - \frac{x}{2} \]

7. **Differentiate \( y \) with respect to \( x \):**
\[ \frac{dy}{dx} = -\frac{1}{2} \]

### Final Answer:
\[ \frac{dy}{dx} = -\frac{1}{2} \]