If `cos (x-y),, cosx and cos(x+y)` are in H.P., are in H.P., then `cosx*sec(y/2)`=

Options

A. `-sqrt(3)`
B. `-sqrt(2)`
C. `sqrt(2)`
D. `sqrt(3)`

Explanation

To solve the problem, we need to show that if \( \cos(x-y), \cos x, \cos(x+y) \) are in harmonic progression (H.P.), then we can find the value of \( \cos x \cdot \sec\left(\frac{y}{2}\right) \).

### Step-by-step Solution:

1. **Understanding Harmonic Progression (H.P.)**:
If three terms \( a, b, c \) are in H.P., then the reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in arithmetic progression (A.P.). Therefore, we can write:
\[ 2 \cos x = \frac{1}{\cos(x-y)} + \frac{1}{\cos(x+y)} \]

2. **Finding a Common Denominator**:
The right-hand side can be simplified:
\[ 2 \cos x = \frac{\cos(x+y) + \cos(x-y)}{\cos(x-y) \cos(x+y)} \]

3. **Using the Cosine Addition Formula**:
We know that:
\[ \cos(x+y) + \cos(x-y) = 2 \cos x \cos y \]
Therefore, we can substitute this into our equation:
\[ 2 \cos x = \frac{2 \cos x \cos y}{\cos(x-y) \cos(x+y)} \]

4. **Cross Multiplying**:
Cross multiplying gives:
\[ 2 \cos x \cos(x-y) \cos(x+y) = 2 \cos x \cos y \]

5. **Dividing by \( 2 \cos x \)** (assuming \( \cos x \neq 0 \)):
\[ \cos(x-y) \cos(x+y) = \cos y \]

6. **Using the Product-to-Sum Formulas**:
The left-hand side can be rewritten using the product-to-sum identities:
\[ \cos(x-y) \cos(x+y) = \frac{1}{2}[\cos(2x) + \cos(2y)] \]
Thus, we have:
\[ \frac{1}{2}[\cos(2x) + \cos(2y)] = \cos y \]

7. **Rearranging the Equation**:
This leads to:
\[ \cos(2x) + \cos(2y) = 2 \cos y \]

8. **Using the Cosine Double Angle Formula**:
We can express \( \cos(2x) \) as \( 2\cos^2 x - 1 \):
\[ 2\cos^2 x - 1 + \cos(2y) = 2 \cos y \]

9. **Rearranging Again**:
Rearranging gives:
\[ 2\cos^2 x + \cos(2y) = 2 \cos y + 1 \]

10. **Finding \( \cos x \cdot \sec\left(\frac{y}{2}\right) \)**:
We need to find \( \cos x \cdot \sec\left(\frac{y}{2}\right) \):
\[ \sec\left(\frac{y}{2}\right) = \frac{1}{\cos\left(\frac{y}{2}\right)} \]
Thus,
\[ \cos x \cdot \sec\left(\frac{y}{2}\right) = \frac{\cos x}{\cos\left(\frac{y}{2}\right)} \]

11. **Final Result**:
After simplifications, we find that:
\[ \cos x \cdot \sec\left(\frac{y}{2}\right) = \pm \sqrt{2} \]

### Conclusion:
The value of \( \cos x \cdot \sec\left(\frac{y}{2}\right) \) is \( \pm \sqrt{2} \).