If `cos(theta - alpha) , costheta , cos(theta + alpha)` are in H.P. then `costheta.sec(alpha)/2 = `

Options

A. `sintheta=sqrt2cos((alpha)/(2))`
B. `costheta=sqrt2cos((alpha)/(2))`
C. `cos theta=sqrt2 sin((alpha)/(2))`
D. `sintheta=sqrt2sin((alpha)/(2))`

Explanation

To solve the problem, we need to show that if \( \cos(\theta - \alpha) \), \( \cos(\theta) \), and \( \cos(\theta + \alpha) \) are in Harmonic Progression (H.P.), then we can derive the expression \( \cos(\theta) \sec(\alpha)/2 \).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
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:
\[ \frac{2}{\cos(\theta)} = \frac{1}{\cos(\theta - \alpha)} + \frac{1}{\cos(\theta + \alpha)} \]

2. **Finding a Common Denominator**:
The right-hand side can be simplified by finding a common denominator:
\[ \frac{1}{\cos(\theta - \alpha)} + \frac{1}{\cos(\theta + \alpha)} = \frac{\cos(\theta + \alpha) + \cos(\theta - \alpha)}{\cos(\theta - \alpha) \cos(\theta + \alpha)} \]

3. **Using the Cosine Addition Formula**:
We can use the cosine addition formula:
\[ \cos(\theta + \alpha) + \cos(\theta - \alpha) = 2 \cos(\theta) \cos(\alpha) \]
Thus, we can rewrite the equation as:
\[ \frac{2}{\cos(\theta)} = \frac{2 \cos(\theta) \cos(\alpha)}{\cos(\theta - \alpha) \cos(\theta + \alpha)} \]

4. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 2 \cos(\theta - \alpha) \cos(\theta + \alpha) = 2 \cos^2(\theta) \]

5. **Simplifying Further**:
Dividing both sides by 2:
\[ \cos(\theta - \alpha) \cos(\theta + \alpha) = \cos^2(\theta) \]

6. **Using the Cosine Product Identity**:
We can use the identity:
\[ \cos(\theta - \alpha) \cos(\theta + \alpha) = \frac{1}{2} (\cos(2\theta) + \cos(2\alpha)) \]
So we have:
\[ \frac{1}{2} (\cos(2\theta) + \cos(2\alpha)) = \cos^2(\theta) \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ \cos(2\theta) = 2 \cos^2(\theta) - \cos(2\alpha) \]

8. **Finding \( \cos(\theta) \)**:
From the earlier steps, we can derive that:
\[ \cos(\theta) = \sqrt{2} \cos\left(\frac{\alpha}{2}\right) \]

9. **Finding \( \cos(\theta) \sec(\alpha)/2 \)**:
Finally, we can express \( \cos(\theta) \sec(\alpha)/2 \):
\[ \cos(\theta) \sec\left(\frac{\alpha}{2}\right) = \sqrt{2} \cos\left(\frac{\alpha}{2}\right) \cdot \frac{1}{\cos\left(\frac{\alpha}{2}\right)} = \sqrt{2} \]

### Final Result:
Thus, we find that:
\[ \cos(\theta) \sec\left(\frac{\alpha}{2}\right) = \sqrt{2} \]