If `piltthetalt 2pi, then sqrt((1+costheta)/(1-costheta))` is equal to

Options

A. `cosectheta+cot theta`
B. `cosectheta-cottheta`
C. `-cosectheta+cot theta`
D. `-cosec theta-cot theta`

Answer

To solve the problem, we need to find the value of the expression \(\sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}\) given that \(\pi < \theta < 2\pi\).

### Step-by-Step Solution:

1. **Define the Expression**:
Let \( f(\theta) = \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} \).

2. **Rationalize the Expression**:
To simplify \( f(\theta) \), we multiply the numerator and denominator by \( 1 + \cos \theta \):
\[ f(\theta) = \sqrt{\frac{(1 + \cos \theta)(1 + \cos \theta)}{(1 - \cos \theta)(1 + \cos \theta)}} \]
This simplifies to:
\[ f(\theta) = \sqrt{\frac{(1 + \cos \theta)^2}{1 - \cos^2 \theta}} \]

3. **Use the Pythagorean Identity**:
We know that \( 1 - \cos^2 \theta = \sin^2 \theta \). Therefore, we can rewrite the expression as:
\[ f(\theta) = \sqrt{\frac{(1 + \cos \theta)^2}{\sin^2 \theta}} \]

4. **Simplify the Square Root**:
The square root of a fraction can be expressed as the fraction of the square roots:
\[ f(\theta) = \frac{1 + \cos \theta}{|\sin \theta|} \]

5. **Determine the Sign of \(\sin \theta\)**:
Since \(\pi < \theta < 2\pi\), \(\sin \theta\) is negative. Therefore, we can express \(|\sin \theta|\) as \(-\sin \theta\):
\[ f(\theta) = \frac{1 + \cos \theta}{-\sin \theta} = -\frac{1 + \cos \theta}{\sin \theta} \]

6. **Express in Terms of Cotangent**:
We can separate the terms:
\[ f(\theta) = -\left(\frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta}\right) = -\csc \theta - \cot \theta \]

### Final Result:
Thus, the expression simplifies to:
\[ f(\theta) = -\csc \theta - \cot \theta \]

### Conclusion:
The final answer is:
\[ \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} = -\csc \theta - \cot \theta \]