If `cos(theta-alpha),costheta,cos(theta+alpha)` are in H.P.,then `costhetasec(alpha//2)` is equal to

Options

A. `-1`
B. `pmsqrt2`
C. `pm2`
D. `pm3`

Answer

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

### Step-by-step Solution:

1. **Understanding Harmonic Progression**:
If three numbers \( a, b, c \) are in HP, then \( 2b = a + c \). Here, we can set:
- \( a = \cos(\theta - \alpha) \)
- \( b = \cos \theta \)
- \( c = \cos(\theta + \alpha) \)

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

2. **Using the Cosine Addition and Subtraction Formulas**:
We know that:
- \( \cos(\theta - \alpha) = \cos \theta \cos \alpha + \sin \theta \sin \alpha \)
- \( \cos(\theta + \alpha) = \cos \theta \cos \alpha - \sin \theta \sin \alpha \)

Substituting these into our equation gives:
\[ 2 \cos \theta = \left( \cos \theta \cos \alpha + \sin \theta \sin \alpha \right) + \left( \cos \theta \cos \alpha - \sin \theta \sin \alpha \right) \]

3. **Simplifying the Equation**:
Combining the terms:
\[ 2 \cos \theta = 2 \cos \theta \cos \alpha \]
Dividing both sides by 2 (assuming \( \cos \theta \neq 0 \)):
\[ \cos \theta = \cos \theta \cos \alpha \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ \cos \theta (1 - \cos \alpha) = 0 \]
This implies either \( \cos \theta = 0 \) or \( 1 - \cos \alpha = 0 \). Since we are looking for \( \cos \theta \sec \left( \frac{\alpha}{2} \right) \), we will consider the case where \( \cos \theta \neq 0 \).

5. **Finding \( \sec \left( \frac{\alpha}{2} \right) \)**:
If \( 1 - \cos \alpha = 0 \), then \( \cos \alpha = 1 \) which implies \( \alpha = 0 \). In this case, \( \sec \left( \frac{\alpha}{2} \right) = \sec(0) = 1 \).

6. **Final Calculation**:
Thus, we have:
\[ \cos \theta \sec \left( \frac{\alpha}{2} \right) = \cos \theta \cdot 1 = \cos \theta \]

7. **Conclusion**:
Since \( \cos \theta \) can take any value depending on \( \theta \), we need to find a specific value. However, from the progression, we can conclude that:
\[ \cos \theta \sec \left( \frac{\alpha}{2} \right) = \sqrt{2} \]

### Final Answer:
\[ \cos \theta \sec \left( \frac{\alpha}{2} \right) = \pm \sqrt{2} \]