Statement:1 if `cos theta=1/7` and `cos phi = 13/14` where `theta` and `phi` both are acute angles, then the value of `theta-phi` is `pi/3`. Statement:2 `cos(pi/3)=1/2`

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-1
B. Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-1
C. Statement-1 is True, Statement-2 is false
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to verify the statements given:

**Statement 1:** If \( \cos \theta = \frac{1}{7} \) and \( \cos \phi = \frac{13}{14} \), where \( \theta \) and \( \phi \) are acute angles, then \( \theta - \phi = \frac{\pi}{3} \).

**Statement 2:** \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

### Step-by-step Solution:

1. **Use the Cosine Difference Identity:**
We know that:
\[ \cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi \]
Given that \( \theta - \phi = \frac{\pi}{3} \), we can write:
\[ \cos(\theta - \phi) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

2. **Substituting Known Values:**
Substitute \( \cos \theta \) and \( \cos \phi \) into the identity:
\[ \frac{1}{2} = \left(\frac{1}{7}\right) \left(\frac{13}{14}\right) + \sin \theta \sin \phi \]

3. **Calculate \( \cos \theta \cos \phi \):**
\[ \cos \theta \cos \phi = \frac{1}{7} \cdot \frac{13}{14} = \frac{13}{98} \]

4. **Rearranging the Equation:**
Substitute \( \cos \theta \cos \phi \) back into the equation:
\[ \frac{1}{2} = \frac{13}{98} + \sin \theta \sin \phi \]
Rearranging gives:
\[ \sin \theta \sin \phi = \frac{1}{2} - \frac{13}{98} \]

5. **Finding a Common Denominator:**
The common denominator for \( \frac{1}{2} \) and \( \frac{13}{98} \) is \( 98 \):
\[ \frac{1}{2} = \frac{49}{98} \]
Thus:
\[ \sin \theta \sin \phi = \frac{49}{98} - \frac{13}{98} = \frac{36}{98} = \frac{18}{49} \]

6. **Finding \( \sin \theta \) and \( \sin \phi \):**
Now we need to find \( \sin \theta \) and \( \sin \phi \):
- For \( \theta \):
\[ \cos^2 \theta + \sin^2 \theta = 1 \implies \sin^2 \theta = 1 - \left(\frac{1}{7}\right)^2 = 1 - \frac{1}{49} = \frac{48}{49} \]
Thus, \( \sin \theta = \frac{\sqrt{48}}{7} = \frac{4\sqrt{3}}{7} \).

- For \( \phi \):
\[ \cos^2 \phi + \sin^2 \phi = 1 \implies \sin^2 \phi = 1 - \left(\frac{13}{14}\right)^2 = 1 - \frac{169}{196} = \frac{27}{196} \]
Thus, \( \sin \phi = \frac{\sqrt{27}}{14} = \frac{3\sqrt{3}}{14} \).

7. **Calculating \( \sin \theta \sin \phi \):**
Now calculate:
\[ \sin \theta \sin \phi = \left(\frac{4\sqrt{3}}{7}\right) \left(\frac{3\sqrt{3}}{14}\right) = \frac{12 \cdot 3}{98} = \frac{36}{98} = \frac{18}{49} \]

8. **Final Verification:**
We have:
\[ \sin \theta \sin \phi = \frac{18}{49} \]
This matches our earlier calculation, confirming that:
\[ \cos(\theta - \phi) = \frac{1}{2} \]
Therefore, Statement 1 is true.

9. **Verification of Statement 2:**
From trigonometric values, we know:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Thus, Statement 2 is also true.

### Conclusion:
Both statements are true, and Statement 2 serves as a correct explanation for Statement 1.