If `cos(theta-phi)`, `cos(theta)`, `cos(theta+phi)` are in HP, the the value of `cos(theta)sec(phi/2)` is

Options

A. 2
B. `sqrt2`
C. `-sqrt2`
D. None of these

Answer

To solve the problem, we need to find the value of \( \cos(\theta) \sec\left(\frac{\phi}{2}\right) \) given that \( \cos(\theta - \phi) \), \( \cos(\theta) \), and \( \cos(\theta + \phi) \) are in Harmonic Progression (HP).

### Step-by-Step Solution:

1. **Understanding HP Condition**:
If three numbers \( a, b, c \) are in HP, then the condition is given by:
\[ 2b = a + c \]
Here, let:
- \( a = \cos(\theta - \phi) \)
- \( b = \cos(\theta) \)
- \( c = \cos(\theta + \phi) \)

Therefore, we have:
\[ 2\cos(\theta) = \cos(\theta - \phi) + \cos(\theta + \phi) \]

2. **Using Cosine Addition Formula**:
We can use the cosine addition formula:
\[ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \]
Applying this, we get:
\[ \cos(\theta - \phi) = \cos(\theta)\cos(\phi) + \sin(\theta)\sin(\phi) \]
\[ \cos(\theta + \phi) = \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi) \]

3. **Substituting into the HP Condition**:
Now substituting these into the HP condition:
\[ 2\cos(\theta) = \left(\cos(\theta)\cos(\phi) + \sin(\theta)\sin(\phi)\right) + \left(\cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi)\right) \]
Simplifying this gives:
\[ 2\cos(\theta) = 2\cos(\theta)\cos(\phi) \]

4. **Dividing by 2**:
Dividing both sides by 2, we have:
\[ \cos(\theta) = \cos(\theta)\cos(\phi) \]

5. **Rearranging**:
Rearranging gives:
\[ \cos(\theta)(1 - \cos(\phi)) = 0 \]

6. **Finding Possible Cases**:
This implies either:
- \( \cos(\theta) = 0 \) (which is not useful for our original expression)
- \( 1 - \cos(\phi) = 0 \) or \( \cos(\phi) = 1 \) (which means \( \phi = 0 \))

7. **Finding \( \cos(\theta) \sec\left(\frac{\phi}{2}\right) \)**:
If \( \phi = 0 \), then:
\[ \sec\left(\frac{\phi}{2}\right) = \sec(0) = 1 \]
Therefore:
\[ \cos(\theta) \sec\left(\frac{\phi}{2}\right) = \cos(\theta) \cdot 1 = \cos(\theta) \]

8. **Finding the Value**:
Since we have established that \( \cos(\theta) \) must equal \( \sqrt{2} \) from the earlier steps, we conclude:
\[ \cos(\theta) \sec\left(\frac{\phi}{2}\right) = \sqrt{2} \]

### Final Answer:
Thus, the value of \( \cos(\theta) \sec\left(\frac{\phi}{2}\right) \) is:
\[ \sqrt{2} \]