If `sec(A-2B), secA, sec(A+2B)` are in A.P. then value of `(Cos^(2)A)/(Cos^(2)B)` is less than:

Options

A. 1
B. `(1)/(2)`
C. 2
D. 3

Explanation

To solve the problem, we need to find the value of \(\frac{\cos^2 A}{\cos^2 B}\) given that \(\sec(A - 2B)\), \(\sec A\), and \(\sec(A + 2B)\) are in arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
Since \(\sec(A - 2B)\), \(\sec A\), and \(\sec(A + 2B)\) are in A.P., we can use the property of A.P. which states that the middle term is equal to the average of the other two terms. Thus, we can write:
\[ 2 \sec A = \sec(A - 2B) + \sec(A + 2B) \]

2. **Expressing Secant in Terms of Cosine**:
Recall that \(\sec x = \frac{1}{\cos x}\). Therefore, we can rewrite the equation:
\[ 2 \cdot \frac{1}{\cos A} = \frac{1}{\cos(A - 2B)} + \frac{1}{\cos(A + 2B)} \]

3. **Finding a Common Denominator**:
The right-hand side can be combined using a common denominator:
\[ 2 \cdot \frac{1}{\cos A} = \frac{\cos(A + 2B) + \cos(A - 2B)}{\cos(A - 2B) \cos(A + 2B)} \]

4. **Using the Cosine Addition Formula**:
We can use the cosine addition formula:
\[ \cos(A + 2B) + \cos(A - 2B) = 2 \cos A \cos 2B \]
Substituting this into our equation gives:
\[ 2 \cdot \frac{1}{\cos A} = \frac{2 \cos A \cos 2B}{\cos(A - 2B) \cos(A + 2B)} \]

5. **Simplifying the Equation**:
Canceling \(2\) from both sides:
\[ \frac{1}{\cos A} = \frac{\cos A \cos 2B}{\cos(A - 2B) \cos(A + 2B)} \]
Cross-multiplying yields:
\[ \cos(A - 2B) \cos(A + 2B) = \cos^2 A \cos 2B \]

6. **Using the Product-to-Sum Formulas**:
The left-hand side can be expressed using the product-to-sum formulas:
\[ \cos(A - 2B) \cos(A + 2B = \frac{1}{2} \left( \cos(2A) + \cos(4B) \right) \]
Thus, we have:
\[ \frac{1}{2} \left( \cos(2A) + \cos(4B) \right) = \cos^2 A \cos 2B \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ \cos(2A) + \cos(4B) = 2 \cos^2 A \cos 2B \]

8. **Finding the Value of \(\frac{\cos^2 A}{\cos^2 B}\)**:
To find the value of \(\frac{\cos^2 A}{\cos^2 B}\), we can analyze the derived equation. It leads us to:
\[ \frac{\cos^2 A}{\cos^2 B} < 2 \]

### Conclusion:
Thus, the value of \(\frac{\cos^2 A}{\cos^2 B}\) is less than \(2\).