The value of `cot^(-1){(sqrt(1-sinx)+sqrt(1+sinx))/(sqrt(1-sinx)-sqrt(1+sinx))}`, where `(pi)/2ltxltpi`, is

Options

A. `pi-x/2`
B. `(pi)/2+x/2`
C. `x/2`
D. `2pi-x/2`

Explanation

To solve the problem, we need to evaluate the expression \( \cot^{-1} \left( \frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}} \right) \) for \( \frac{\pi}{2} < x < \pi \).

### Step 1: Define the expression
Let
\[ y = \frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}} \]

### Step 2: Rationalize the denominator
To simplify \( y \), we can multiply the numerator and denominator by the conjugate of the denominator:
\[ y = \frac{(\sqrt{1 - \sin x} + \sqrt{1 + \sin x})(\sqrt{1 - \sin x} + \sqrt{1 + \sin x})}{(\sqrt{1 - \sin x} - \sqrt{1 + \sin x})(\sqrt{1 - \sin x} + \sqrt{1 + \sin x})} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ (\sqrt{1 - \sin x})^2 - (\sqrt{1 + \sin x})^2 = (1 - \sin x) - (1 + \sin x) = -2\sin x \]

### Step 4: Simplify the numerator
The numerator becomes:
\[ (\sqrt{1 - \sin x} + \sqrt{1 + \sin x})^2 = (1 - \sin x) + (1 + \sin x) + 2\sqrt{(1 - \sin x)(1 + \sin x)} = 2 + 2\sqrt{1 - \sin^2 x} = 2 + 2\cos x \]

### Step 5: Combine results
Now we can write \( y \) as:
\[ y = \frac{2 + 2\cos x}{-2\sin x} = -\frac{1 + \cos x}{\sin x} \]

### Step 6: Express in terms of cotangent
Notice that:
\[ -\frac{1 + \cos x}{\sin x} = -\cot\left(\frac{x}{2}\right) \]
This is because \( 1 + \cos x = 2\cos^2\left(\frac{x}{2}\right) \) and \( \sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) \).

### Step 7: Find cotangent inverse
Thus, we have:
\[ y = -\cot\left(\frac{x}{2}\right) \]
Therefore,
\[ \cot^{-1}(y) = \cot^{-1}(-\cot\left(\frac{x}{2}\right)) \]

### Step 8: Use the cotangent identity
Using the identity \( \cot^{-1}(-\cot \theta) = \pi - \theta \) for \( \theta \) in the appropriate range, we get:
\[ \cot^{-1}(-\cot\left(\frac{x}{2}\right)) = \pi - \frac{x}{2} \]

### Final Answer
Thus, the value of the expression is:
\[ \cot^{-1} \left( \frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}} \right) = \pi - \frac{x}{2} \]