If `(5pi)/(2)ltxlt3pi`, then the value of the expression `(sqrt(1-sinx)+sqrt(1+sinx))/(sqrt(1-sinx)-sqrt(1+sinx))` is

Options

A. `-cot(x/2)`
B. `cot (x/2)`
C. `tan(x/2)`
D. `-tan(x/2)`

Answer

To solve the expression \(\frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}\) given that \(\frac{5\pi}{2} < x < 3\pi\), we will follow these steps:

### Step 1: Rationalize the Expression
We start by rationalizing the denominator. We multiply the numerator and the denominator by \(\sqrt{1 - \sin x} + \sqrt{1 + \sin x}\):

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

### Step 2: Simplify the Numerator
The numerator becomes:

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

This simplifies to:

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

### Step 3: Simplify the Denominator
The denominator becomes:

\[ (1 - \sin x) - (1 + \sin x) = -2\sin x \]

### Step 4: Combine the Results
Now we can rewrite the expression:

\[ \frac{2 + 2\cos x}{-2\sin x} \]

### Step 5: Factor Out Common Terms
Factoring out the common terms gives us:

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

### Step 6: Use Trigonometric Identities
Using the identity \(1 + \cos x = 2\cos^2\left(\frac{x}{2}\right)\) and \(\sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)\), we can rewrite:

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

### Final Answer
Thus, the value of the expression is:

\[ -\cot\left(\frac{x}{2}\right) \]