What is the derivative of `sqrt((1+cosx)/(1-cosx))`?

Options

A. `(1)/(2)sec^(2).(x)/(2)`
B. `-(1)/(2)"cosec^(2)(x)/(2)`
C. `-"cosec"^(2).(x)/(2)`
D. None of these

Explanation

To find the derivative of the function \( y = \sqrt{\frac{1 + \cos x}{1 - \cos x}} \), we will follow these steps:

### Step 1: Rewrite the function using trigonometric identities
We can use the identities:
- \( 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \)
- \( 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \)

Thus, we can rewrite \( y \) as:
\[ y = \sqrt{\frac{2 \cos^2\left(\frac{x}{2}\right)}{2 \sin^2\left(\frac{x}{2}\right)}} \]
This simplifies to:
\[ y = \sqrt{\frac{\cos^2\left(\frac{x}{2}\right)}{\sin^2\left(\frac{x}{2}\right)}} = \frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} = \cot\left(\frac{x}{2}\right) \]

### Step 2: Differentiate the function
Now, we need to differentiate \( y = \cot\left(\frac{x}{2}\right) \). The derivative of \( \cot u \) is \( -\csc^2 u \) where \( u = \frac{x}{2} \).

Using the chain rule:
\[ \frac{dy}{dx} = -\csc^2\left(\frac{x}{2}\right) \cdot \frac{d}{dx}\left(\frac{x}{2}\right) = -\csc^2\left(\frac{x}{2}\right) \cdot \frac{1}{2} \]

Thus, we have:
\[ \frac{dy}{dx} = -\frac{1}{2} \csc^2\left(\frac{x}{2}\right) \]

### Step 3: Final expression
The final expression for the derivative is:
\[ \frac{dy}{dx} = -\frac{1}{2} \csc^2\left(\frac{x}{2}\right) \]