`(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))=?` (x is in IV quadrant)

Options

A. `(x)/(2)`
B. `tan""(x)/(2)`
C. `sec""(x)/(2)`
D. `cosec""(x)/(2)`

Answer

To solve the expression \(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\) where \(x\) is in the fourth quadrant, we will follow these steps:

### Step 1: Rationalize the Denominator
We start by multiplying the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{1+\sin x} + \sqrt{1-\sin x}\).

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

### Step 2: Apply the Difference of Squares
The denominator simplifies using the difference of squares formula \(a^2 - b^2\):

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

### Step 3: Expand the Numerator
Now we expand the numerator:

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

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

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

### Step 5: Simplify the Expression
We can factor out a 2 from the numerator:

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

### Step 6: Recognize the Trigonometric Identity
The expression \(\frac{1 + \cos x}{\sin x}\) can be rewritten using the cotangent function:

\[ \frac{1 + \cos x}{\sin x} = \cot\left(\frac{x}{2}\right) \]

### Final Answer
Thus, the final answer is:

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