If `sin alpha, sin beta, sin gamma` are in `AP` and `cos alpha, cos beta, cos gamma` are in `GP`, then the value of
`(cos^(2)alpha+cos^(2)gamma+4 cos alpha cos gamma-2 sin alpha sin gamma-2)/(1-2 sin^(2)beta)`, where `beta != (pi)/(4)`, is equal to

Answer

To solve the problem step by step, we start by analyzing the conditions given in the question.

### Step 1: Understanding the conditions
We know that \( \sin \alpha, \sin \beta, \sin \gamma \) are in Arithmetic Progression (AP) and \( \cos \alpha, \cos \beta, \cos \gamma \) are in Geometric Progression (GP).

### Step 2: Using the AP condition
From the property of AP, we have:
\[ \sin \beta - \sin \alpha = \sin \gamma - \sin \beta \]
Rearranging gives:
\[ 2 \sin \beta = \sin \alpha + \sin \gamma \]

### Step 3: Squaring both sides
Now, squaring both sides:
\[ (2 \sin \beta)^2 = (\sin \alpha + \sin \gamma)^2 \]
This simplifies to:
\[ 4 \sin^2 \beta = \sin^2 \alpha + \sin^2 \gamma + 2 \sin \alpha \sin \gamma \]

### Step 4: Using the identity for sine
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can rewrite:
\[ 4 \sin^2 \beta = (1 - \cos^2 \alpha) + (1 - \cos^2 \gamma) + 2 \sin \alpha \sin \gamma \]
This leads to:
\[ 4 \sin^2 \beta = 2 - \cos^2 \alpha - \cos^2 \gamma + 2 \sin \alpha \sin \gamma \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ \cos^2 \alpha + \cos^2 \gamma + 2 \sin \alpha \sin \gamma = 2 - 4 \sin^2 \beta \]

### Step 6: Using the GP condition
From the GP condition \( \cos \beta^2 = \cos \alpha \cos \gamma \), we can express:
\[ \cos^2 \beta = \cos \alpha \cos \gamma \]

### Step 7: Setting up the expression to evaluate
We need to evaluate:
\[ \frac{\cos^2 \alpha + \cos^2 \gamma + 4 \cos \alpha \cos \gamma - 2 \sin \alpha \sin \gamma - 2}{1 - 2 \sin^2 \beta} \]

### Step 8: Substituting values
Substituting from our earlier results:
\[ \cos^2 \alpha + \cos^2 \gamma + 4 \cos \alpha \cos \gamma = 2 - 4 \sin^2 \beta + 4 \cos \beta^2 \]
Thus, the numerator becomes:
\[ (2 - 4 \sin^2 \beta + 4 \cos^2 \beta - 2 \sin \alpha \sin \gamma - 2) \]

### Step 9: Simplifying the expression
Now, simplifying the numerator:
\[ = 4 \cos^2 \beta - 4 \sin^2 \beta = 4(\cos^2 \beta - \sin^2 \beta) \]

### Step 10: Final evaluation
The denominator simplifies to:
\[ 1 - 2 \sin^2 \beta = \cos 2\beta \]
Thus, the expression simplifies to:
\[ \frac{4(\cos^2 \beta - \sin^2 \beta)}{\cos 2\beta} = 4 \]

### Conclusion
Therefore, the value of the expression is:
\[ \boxed{4} \]