In an actue-angled triangle ABC
Statement-1: `tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2)ge1`
Statement-2: `tanAtanB tanCge3sqrt3`

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 problem, we need to analyze both statements given in the acute-angled triangle ABC.

### Step 1: Understanding the Triangle
In an acute-angled triangle ABC, we know that:
- \( A + B + C = \pi \) (in radians).

### Step 2: Analyze Statement 1
Statement 1:
\[ \frac{\tan^2(A/2)}{2} + \frac{\tan^2(B/2)}{2} + \frac{\tan^2(C/2)}{2} \geq 1 \]
Let \( x = \tan(A/2) \), \( y = \tan(B/2) \), and \( z = \tan(C/2) \). Then we can rewrite the statement as:
\[ \frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} \geq 1 \]
Multiplying through by 2 gives:
\[ x^2 + y^2 + z^2 \geq 2 \]

### Step 3: Using the Identity
From the identity for angles in a triangle, we have:
\[ x^2 + y^2 + z^2 = (x+y+z)^2 - 2(xy + yz + zx) \]
Using the relation \( xy + yz + zx = 1 \) (derived from the tangent addition formulas), we can substitute:
\[ x^2 + y^2 + z^2 = (x+y+z)^2 - 2 \]
Thus, we need to show:
\[ (x+y+z)^2 - 2 \geq 2 \implies (x+y+z)^2 \geq 4 \]
This is true since \( x, y, z \) are all positive in an acute triangle.

### Step 4: Analyze Statement 2
Statement 2:
\[ \tan A \tan B \tan C \geq 3\sqrt{3} \]
Using the identity for the product of tangents in a triangle, we know:
\[ \tan A \tan B \tan C = \frac{\tan A + \tan B + \tan C}{\tan A + \tan B + \tan C - 1} \]
For an acute triangle, this inequality holds true.

### Conclusion
Both statements are true, but we need to check if one is a correct explanation for the other. Since we derived each statement independently and they do not rely on one another, we conclude:

- **Statement 1** is true.
- **Statement 2** is true.

However, since they are independent, we cannot say that one explains the other.

### Final Answer
Thus, the correct answer is that both statements are true, but Statement 2 does not explain Statement 1.

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