Statement I In `a DeltaABC, sum (cos ^(2)""(A)/(2))/(a )` has the value equal to `(s^(2))/(abc).`
Statement II in `a Delta ABC, cos ""A/2=sqrt(((s-b)(s-c))/(bc))`
`cos ""(beta)/(2)=sqrt(((s-a) (s-c))/(ac)), cos ""c/2= sqrt(((s-a)(s-b))/(ab))`

Options

A. Statement I is True, Statement II is True, Statement II is a correct explanation for Statement I.
B. Statement I is True, Statement II is True, Statement II is NOT a correct explanation for Statement I.
C. Statement I is True, Statement II is False.
D. Statement I is False, Statement II is True.

Answer

To solve the given problem, we will analyze both statements step by step.

### Step 1: Analyze Statement I
Statement I claims that:
\[ \sum \frac{\cos^2 \left(\frac{A}{2}\right)}{a} = \frac{s^2}{abc} \]

Where \( s \) is the semi-perimeter of triangle \( ABC \).

### Step 2: Use the formula for \(\cos \frac{A}{2}\)
The formula for \(\cos \frac{A}{2}\) is given by:
\[ \cos \frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}} \]
Thus,
\[ \cos^2 \frac{A}{2} = \frac{s(s-a)}{bc} \]

### Step 3: Substitute into the summation
Now, substituting this into the summation:
\[ \sum \frac{\cos^2 \left(\frac{A}{2}\right)}{a} = \frac{s(s-a)}{abc} + \frac{s(s-b)}{abc} + \frac{s(s-c)}{abc} \]

### Step 4: Combine the terms
Combining these terms gives:
\[ \sum \frac{\cos^2 \left(\frac{A}{2}\right)}{a} = \frac{s}{abc} \left( (s-a) + (s-b) + (s-c) \right) \]

### Step 5: Simplify the expression
Now, simplifying the expression inside the parentheses:
\[ (s-a) + (s-b) + (s-c) = 3s - (a+b+c) = 3s - 2s = s \]
Thus, we have:
\[ \sum \frac{\cos^2 \left(\frac{A}{2}\right)}{a} = \frac{s \cdot s}{abc} = \frac{s^2}{abc} \]

### Conclusion for Statement I
Therefore, Statement I is true:
\[ \sum \frac{\cos^2 \left(\frac{A}{2}\right)}{a} = \frac{s^2}{abc} \]

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### Step 6: Analyze Statement II
Statement II claims:
\[ \cos \frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}}, \quad \cos \frac{B}{2} = \sqrt{\frac{(s-a)(s-c)}{ac}}, \quad \cos \frac{C}{2} = \sqrt{\frac{(s-a)(s-b)}{ab}} \]

### Step 7: Verify the formulas
We already know the correct formula for \(\cos \frac{A}{2}\):
\[ \cos \frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}} \]
The formulas provided in Statement II do not match the known formula for \(\cos \frac{A}{2}\), \(\cos \frac{B}{2}\), and \(\cos \frac{C}{2}\).

### Conclusion for Statement II
Thus, Statement II is false.

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### Final Answer
- Statement I is true.
- Statement II is false.

### Options
The correct option is that Statement 1 is true and Statement 2 is false.

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