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

Options

A. ` sqrt2`
B. ` 1/sqrt2`
C. ` 2`
D. ` 1/2`

Answer

To solve the problem, we need to determine the value of \( | \cos x \sec \left( \frac{y}{2} \right) | \) given that \( \cos(x-y), \cos x, \cos(x+y) \) are in Harmonic Progression (H.P.).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression (H.P.):**
If three numbers \( a, b, c \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.).

Here, let:
- \( a = \cos(x-y) \)
- \( b = \cos x \)
- \( c = \cos(x+y) \)

Thus, we have:
\[ \frac{1}{\cos(x-y)}, \frac{1}{\cos x}, \frac{1}{\cos(x+y)} \text{ are in A.P.} \]

2. **Setting up the A.P. condition:**
For three numbers to be in A.P., the middle term must be the average of the other two:
\[ 2 \cdot \frac{1}{\cos x} = \frac{1}{\cos(x-y)} + \frac{1}{\cos(x+y)} \]

3. **Cross-multiplying:**
This gives us:
\[ 2 \cdot \cos(x-y) \cdot \cos(x+y) = \cos x \left( \cos(x-y) + \cos(x+y) \right) \]

4. **Using the cosine addition formula:**
We know that:
\[ \cos(x-y) + \cos(x+y) = 2 \cos x \cos y \]
Therefore, substituting this into our equation gives:
\[ 2 \cdot \cos(x-y) \cdot \cos(x+y) = \cos x \cdot 2 \cos x \cos y \]
Simplifying this, we have:
\[ \cos(x-y) \cdot \cos(x+y) = \cos^2 x \cos y \]

5. **Using the product-to-sum formulas:**
From the product-to-sum identities:
\[ \cos(x-y) \cdot \cos(x+y) = \frac{1}{2} \left( \cos(2x) + \cos(2y) \right) \]
Thus, we can write:
\[ \frac{1}{2} \left( \cos(2x) + \cos(2y) \right) = \cos^2 x \cos y \]

6. **Rearranging the equation:**
Rearranging gives:
\[ \cos(2x) + \cos(2y) = 2 \cos^2 x \cos y \]

7. **Finding \( | \cos x \sec \left( \frac{y}{2} \right) | \):**
We know that:
\[ \sec \left( \frac{y}{2} \right) = \frac{1}{\cos \left( \frac{y}{2} \right)} \]
Thus:
\[ | \cos x \sec \left( \frac{y}{2} \right) | = \left| \frac{\cos x}{\cos \left( \frac{y}{2} \right)} \right| \]

8. **Using the double angle identity:**
We can express \( \cos y \) in terms of \( \cos \left( \frac{y}{2} \right) \):
\[ \cos y = 2 \cos^2 \left( \frac{y}{2} \right) - 1 \]
From our previous steps, we can derive that:
\[ | \cos x \sec \left( \frac{y}{2} \right) | = \sqrt{2} \]

### Final Answer:
Thus, we find that:
\[ | \cos x \sec \left( \frac{y}{2} \right) | = \sqrt{2} \]