`|(1,cos(pi//12),cos(pi//3)),(cos(pi//12),1,cos(pi//4)),(cos(pi//3),cos(pi//4),1)|=1`_________________

Answer

To solve the determinant

\[ D = \begin{vmatrix} 1 & \cos(\frac{\pi}{12}) & \cos(\frac{\pi}{3}) \\ \cos(\frac{\pi}{12}) & 1 & \cos(\frac{\pi}{4}) \\ \cos(\frac{\pi}{3}) & \cos(\frac{\pi}{4}) & 1 \end{vmatrix} \]

we will use the properties of determinants and trigonometric identities.

### Step 1: Write the determinant
We start by writing the determinant explicitly:

\[ D = 1 \cdot \begin{vmatrix} 1 & \cos(\frac{\pi}{4}) \\ \cos(\frac{\pi}{4}) & 1 \end{vmatrix} - \cos(\frac{\pi}{12}) \cdot \begin{vmatrix} \cos(\frac{\pi}{12}) & \cos(\frac{\pi}{4}) \\ \cos(\frac{\pi}{3}) & 1 \end{vmatrix} + \cos(\frac{\pi}{3}) \cdot \begin{vmatrix} \cos(\frac{\pi}{12}) & 1 \\ \cos(\frac{\pi}{3}) & \cos(\frac{\pi}{4}) \end{vmatrix} \]

### Step 2: Calculate the 2x2 determinants
1. Calculate the first 2x2 determinant:

\[ \begin{vmatrix} 1 & \cos(\frac{\pi}{4}) \\ \cos(\frac{\pi}{4}) & 1 \end{vmatrix} = 1 \cdot 1 - \cos^2(\frac{\pi}{4}) = 1 - \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{1}{2} = \frac{1}{2} \]

2. Calculate the second 2x2 determinant:

\[ \begin{vmatrix} \cos(\frac{\pi}{12}) & \cos(\frac{\pi}{4}) \\ \cos(\frac{\pi}{3}) & 1 \end{vmatrix} = \cos(\frac{\pi}{12}) \cdot 1 - \cos(\frac{\pi}{4}) \cdot \cos(\frac{\pi}{3}) = \cos(\frac{\pi}{12}) - \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right) = \cos(\frac{\pi}{12}) - \frac{1}{2\sqrt{2}} \]

3. Calculate the third 2x2 determinant:

\[ \begin{vmatrix} \cos(\frac{\pi}{12}) & 1 \\ \cos(\frac{\pi}{3}) & \cos(\frac{\pi}{4}) \end{vmatrix} = \cos(\frac{\pi}{12}) \cdot \cos(\frac{\pi}{4}) - 1 \cdot \cos(\frac{\pi}{3}) = \frac{\cos(\frac{\pi}{12})}{\sqrt{2}} - \frac{1}{2} \]

### Step 3: Substitute back into the determinant
Now substituting back into the expression for \(D\):

\[ D = 1 \cdot \frac{1}{2} - \cos(\frac{\pi}{12}) \left(\cos(\frac{\pi}{12}) - \frac{1}{2\sqrt{2}}\right) + \frac{1}{2} \left(\frac{\cos(\frac{\pi}{12})}{\sqrt{2}} - \frac{1}{2}\right) \]

### Step 4: Simplify the expression
1. The first term is \( \frac{1}{2} \).
2. The second term expands to:

\[ -\cos^2(\frac{\pi}{12}) + \frac{\cos(\frac{\pi}{12})}{2\sqrt{2}} \]

3. The third term expands to:

\[ \frac{1}{2\sqrt{2}} \cos(\frac{\pi}{12}) - \frac{1}{4} \]

Combining all these terms, we get:

\[ D = \frac{1}{2} - \cos^2(\frac{\pi}{12}) + \frac{1}{2\sqrt{2}} \cos(\frac{\pi}{12}) + \frac{1}{2\sqrt{2}} \cos(\frac{\pi}{12}) - \frac{1}{4} \]

### Step 5: Final simplification
Combine like terms:

\[ D = \frac{1}{4} - \cos^2(\frac{\pi}{12}) + \frac{1}{\sqrt{2}} \cos(\frac{\pi}{12}) \]

Setting \(D = 1\) leads us to solve for \(\cos(\frac{\pi}{12})\) using the quadratic equation formed.