`cos^(-1)(sqrt(1+cos x)/2)`

Answer

To solve the problem \( y = \cos^{-1}\left(\frac{\sqrt{1 + \cos x}}{2}\right) \), we will follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
We know from trigonometric identities that:
\[ \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \]
This means we can express \( \frac{\sqrt{1 + \cos x}}{2} \) in terms of cosine.

### Step 2: Identify the angle
From the identity, we can set:
\[ \frac{\sqrt{1 + \cos x}}{2} = \cos \theta \]
This implies:
\[ \sqrt{1 + \cos x} = 2 \cos \theta \]
Squaring both sides, we get:
\[ 1 + \cos x = 4 \cos^2 \theta \]

### Step 3: Solve for \( \theta \)
From the equation \( 1 + \cos x = 4 \cos^2 \theta \), we can express \( \cos \theta \) as:
\[ \cos^2 \theta = \frac{1 + \cos x}{4} \]
Thus:
\[ \theta = \cos^{-1}\left(\sqrt{\frac{1 + \cos x}{4}}\right) \]

### Step 4: Relate \( y \) and \( x \)
Since \( y = \cos^{-1}(\cos \theta) \), we have:
\[ y = \theta \]
This means:
\[ y = \cos^{-1}\left(\sqrt{\frac{1 + \cos x}{4}}\right) \]

### Step 5: Differentiate \( y \) with respect to \( x \)
To differentiate \( y \), we use the chain rule. The derivative of \( \cos^{-1}(u) \) is:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = \frac{\sqrt{1 + \cos x}}{2} \).

### Step 6: Find \( \frac{du}{dx} \)
First, we find \( u \):
\[ u = \frac{\sqrt{1 + \cos x}}{2} \]
Now, differentiating \( u \):
\[ \frac{du}{dx} = \frac{1}{2\sqrt{1 + \cos x}} \cdot (-\sin x) \]
Thus:
\[ \frac{du}{dx} = -\frac{\sin x}{2\sqrt{1 + \cos x}} \]

### Step 7: Substitute \( u \) and \( \frac{du}{dx} \) into the derivative
Now substituting back into the derivative:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - \left(\frac{\sqrt{1 + \cos x}}{2}\right)^2}} \cdot \left(-\frac{\sin x}{2\sqrt{1 + \cos x}}\right) \]

### Step 8: Simplify the expression
We simplify \( \sqrt{1 - \left(\frac{\sqrt{1 + \cos x}}{2}\right)^2} \):
\[ \sqrt{1 - \frac{1 + \cos x}{4}} = \sqrt{\frac{4 - (1 + \cos x)}{4}} = \sqrt{\frac{3 - \cos x}{4}} = \frac{\sqrt{3 - \cos x}}{2} \]

### Final Step: Combine everything
Thus, the derivative becomes:
\[ \frac{dy}{dx} = \frac{\sin x}{\sqrt{3 - \cos x} \cdot \sqrt{1 + \cos x}} \]

### Final Answer
\[ \frac{dy}{dx} = \frac{\sin x}{\sqrt{3 - \cos x} \cdot \sqrt{1 + \cos x}} \]