`logsqrt((1-cos x)/(1+cos x))`

Answer

To differentiate the expression \( y = \log \sqrt{\frac{1 - \cos x}{1 + \cos x}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \log \sqrt{\frac{1 - \cos x}{1 + \cos x}} \]
Using the property of logarithms, we can rewrite this as:
\[ y = \frac{1}{2} \log \left( \frac{1 - \cos x}{1 + \cos x} \right) \]

### Step 2: Use trigonometric identities
We can utilize the trigonometric identities:
\[ 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \quad \text{and} \quad 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \]
Substituting these into our expression gives:
\[ y = \frac{1}{2} \log \left( \frac{2 \sin^2\left(\frac{x}{2}\right)}{2 \cos^2\left(\frac{x}{2}\right)} \right) \]
This simplifies to:
\[ y = \frac{1}{2} \log \left( \tan^2\left(\frac{x}{2}\right) \right) \]

### Step 3: Simplify further
Using the property of logarithms, we can simplify this further:
\[ y = \log \left( \tan\left(\frac{x}{2}\right) \right) \]

### Step 4: Differentiate with respect to \( x \)
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{\tan\left(\frac{x}{2}\right)} \cdot \frac{d}{dx}\left(\tan\left(\frac{x}{2}\right)\right) \]
Using the chain rule:
\[ \frac{d}{dx}\left(\tan\left(\frac{x}{2}\right)\right) = \sec^2\left(\frac{x}{2}\right) \cdot \frac{1}{2} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{\tan\left(\frac{x}{2}\right)} \cdot \sec^2\left(\frac{x}{2}\right) \cdot \frac{1}{2} \]

### Step 5: Simplify the derivative
We can express \( \sec^2\left(\frac{x}{2}\right) \) in terms of sine and cosine:
\[ \frac{dy}{dx} = \frac{1}{\tan\left(\frac{x}{2}\right)} \cdot \frac{1}{\cos^2\left(\frac{x}{2}\right)} \cdot \frac{1}{2} \]
Since \( \tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} \), we can rewrite:
\[ \frac{dy}{dx} = \frac{1}{\frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)}} \cdot \frac{1}{\cos^2\left(\frac{x}{2}\right)} \cdot \frac{1}{2} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} \cdot \frac{1}{\cos^2\left(\frac{x}{2}\right)} \cdot \frac{1}{2} = \frac{1}{2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right)} \]
Using the identity \( 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = \sin(x) \):
\[ \frac{dy}{dx} = \frac{1}{\sin(x)} \]

### Final Result
Thus, the derivative of the given expression is:
\[ \frac{dy}{dx} = \frac{1}{\sin x} \]