Statement I In a `Delta ABC, if cos ^(2)""A/2 + cos ^(2)""B/2 +cos ^(2) ""C/2=y(x^(2)+(1)/(x^(2)))` then the maximum value of `y is 9/8.`
Statement II In a `Delta 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 one by one.

### Step 1: Analyze Statement I
The first statement is:
\[ \cos^2\left(\frac{A}{2}\right) + \cos^2\left(\frac{B}{2}\right) + \cos^2\left(\frac{C}{2}\right) = y\left(x^2 + \frac{1}{x^2}\right) \]
We need to find the maximum value of \(y\).

#### Step 1.1: Use Known Results
From properties of triangles, we know:
\[ \cos^2\left(\frac{A}{2}\right) + \cos^2\left(\frac{B}{2}\right) + \cos^2\left(\frac{C}{2}\right) \leq \frac{9}{4} \]
This will be our Equation (1).

#### Step 1.2: Analyze \(x^2 + \frac{1}{x^2}\)
We can express \(x^2 + \frac{1}{x^2}\) as:
\[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \]
This implies that \(x^2 + \frac{1}{x^2} \geq 2\) (Equation 2).

#### Step 1.3: Combine Results
From Equation (1) and Equation (2), we can substitute:
\[ y\left(x^2 + \frac{1}{x^2}\right) \leq \frac{9}{4} \]
Substituting the minimum value of \(x^2 + \frac{1}{x^2}\) (which is 2):
\[ y \cdot 2 \leq \frac{9}{4} \]
This leads to:
\[ y \leq \frac{9}{8} \]
Thus, the maximum value of \(y\) is \(\frac{9}{8}\).

### Conclusion for Statement I
The first statement is correct.

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### Step 2: Analyze Statement II
The second statement is:
\[ \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{1}{8} \]
We need to prove this statement.

#### Step 2.1: Use the AM-GM Inequality
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \sqrt[3]{\sin^2\left(\frac{A}{2}\right) \cdot \sin^2\left(\frac{B}{2}\right) \cdot \sin^2\left(\frac{C}{2}\right)} \leq \frac{\sin^2\left(\frac{A}{2}\right) + \sin^2\left(\frac{B}{2}\right) + \sin^2\left(\frac{C}{2}\right)}{3} \]

#### Step 2.2: Use Known Results
We know:
\[ \sin^2\left(\frac{A}{2}\right) + \sin^2\left(\frac{B}{2}\right) + \sin^2\left(\frac{C}{2}\right) \geq \frac{3}{4} \]
This will be our Equation (3).

#### Step 2.3: Substitute in AM-GM
Substituting Equation (3) into the AM-GM inequality gives:
\[ \sqrt[3]{\sin^2\left(\frac{A}{2}\right) \cdot \sin^2\left(\frac{B}{2}\right) \cdot \sin^2\left(\frac{C}{2}\right)} \leq \frac{\frac{3}{4}}{3} = \frac{1}{4} \]
Cubing both sides:
\[ \sin^2\left(\frac{A}{2}\right) \cdot \sin^2\left(\frac{B}{2}\right) \cdot \sin^2\left(\frac{C}{2}\right) \leq \left(\frac{1}{4}\right)^3 = \frac{1}{64} \]

#### Step 2.4: Relate to Original Statement
Taking the square root gives:
\[ \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{1}{8} \]

### Conclusion for Statement II
The second statement is also correct.

### Final Conclusion
Both statements are correct.

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