Statement I In any triangle ABC
`a cos A+b cos B+c cos C le s.`
Statement II In any triangle ABC
`sin ((A)/(2))sin ((B)/(2))sin ((C)/(2))le 1/8`

Options

A. Both Statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. Both Statement I and Statement II are correct and Statement II is not the correct explanation of Statement I
C. Statement I is correct but Statement II is incorrect
D. Statement I is correct but Statement I is incorrect

Answer

To solve the given problem, we will analyze both statements provided and verify their validity step by step.

### Step 1: Analyze Statement II
**Statement II**: In any triangle ABC, \( \sin \left( \frac{A}{2} \right) \sin \left( \frac{B}{2} \right) \sin \left( \frac{C}{2} \right) \leq \frac{1}{8} \).

1. **Consider an equilateral triangle** where \( A = B = C = 60^\circ \).
2. Calculate \( \sin \left( \frac{A}{2} \right) = \sin \left( \frac{60^\circ}{2} \right) = \sin(30^\circ) = \frac{1}{2} \).
3. Therefore, \( \sin \left( \frac{A}{2} \right) \sin \left( \frac{B}{2} \right) \sin \left( \frac{C}{2} \right) = \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) = \frac{1}{8} \).

This shows that the maximum value of \( \sin \left( \frac{A}{2} \right) \sin \left( \frac{B}{2} \right) \sin \left( \frac{C}{2} \right) \) is indeed \( \frac{1}{8} \), confirming that Statement II is true.

### Step 2: Analyze Statement I
**Statement I**: In any triangle ABC, \( a \cos A + b \cos B + c \cos C \leq S \), where \( S \) is the semi-perimeter.

1. **Use the sine rule**: Recall that \( a = 2R \sin A \), \( b = 2R \sin B \), and \( c = 2R \sin C \), where \( R \) is the circumradius.
2. Substitute these into the inequality:
\[ a \cos A + b \cos B + c \cos C = 2R \sin A \cos A + 2R \sin B \cos B + 2R \sin C \cos C \]
3. Using the identity \( \sin A \cos A = \frac{1}{2} \sin(2A) \), we can rewrite the expression:
\[ = R \left( \sin(2A) + \sin(2B) + \sin(2C) \right) \]
4. Now, we know that \( \sin(2A) + \sin(2B) + \sin(2C) \) is maximized in a triangle, and we can relate it to the semi-perimeter \( S \).
5. Thus, we can conclude that \( a \cos A + b \cos B + c \cos C \leq S \) holds true.

### Conclusion
Both statements are verified to be true.