Statement-1: If `tanA, tanB` are the roots of the equation `x^(2)-ax-1=0,then sin^(2)(A+B)=(a^(2))/(1+a^(2))`
Statement-2: `sin^(2)(A+B)=(tan^(2)(A+B))/(1+tan^(2)(A+B))`

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 explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

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

### Step 1: Analyze Statement 1
**Statement 1:** If \( \tan A, \tan B \) are the roots of the equation \( x^2 - ax - 1 = 0 \), then \( \sin^2(A+B) = \frac{a^2}{1 + a^2} \).

1. **Identify the roots:** From the quadratic equation \( x^2 - ax - 1 = 0 \), we can identify the sum and product of the roots:
- Sum of roots \( \tan A + \tan B = a \)
- Product of roots \( \tan A \tan B = -1 \)

### Step 2: Use the tangent addition formula
2. **Tangent of sum formula:** We know that:
\[ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Substituting the values we have:
\[ \tan(A+B) = \frac{a}{1 - (-1)} = \frac{a}{2} \]

### Step 3: Calculate \( \sin^2(A+B) \)
3. **Relate \( \sin^2(A+B) \) to \( \tan(A+B) \):**
\[ \sin^2(A+B) = \frac{\tan^2(A+B)}{1 + \tan^2(A+B)} \]
Substituting \( \tan(A+B) = \frac{a}{2} \):
\[ \tan^2(A+B) = \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} \]
Therefore,
\[ \sin^2(A+B) = \frac{\frac{a^2}{4}}{1 + \frac{a^2}{4}} = \frac{\frac{a^2}{4}}{\frac{4 + a^2}{4}} = \frac{a^2}{4 + a^2} \]

### Step 4: Compare with Statement 1
4. **Check Statement 1:** We found that:
\[ \sin^2(A+B) = \frac{a^2}{4 + a^2} \]
This does not match with \( \frac{a^2}{1 + a^2} \). Hence, **Statement 1 is false**.

### Step 5: Analyze Statement 2
**Statement 2:** \( \sin^2(A+B) = \frac{\tan^2(A+B)}{1 + \tan^2(A+B)} \)

5. **Verification of Statement 2:** We have already derived that:
\[ \sin^2(A+B) = \frac{\tan^2(A+B)}{1 + \tan^2(A+B)} \]
This is a known trigonometric identity, and since we derived it correctly, **Statement 2 is true**.

### Conclusion
- **Statement 1:** False
- **Statement 2:** True

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

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