Statement-1: In an acute angled triangle minimum value of `tan alpha + tanbeta + tan gamma` is `3sqrt(3)`. And Statement-2: If a,b,c are three positive real numbers then `(a+b+c)/3 ge sqrt(abc)` into in a `triangleABC`, `tanA+ tanB + tanC= tanA. tanB.tanC`

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-8
B. Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-8
C. Statement-1 is True, Statement-2 is false
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze both statements and derive the necessary conclusions step by step.

### Step 1: Analyze Statement 1
We need to find the minimum value of \( \tan A + \tan B + \tan C \) in an acute-angled triangle \( ABC \).

**Given:**
- In triangle \( ABC \), \( A + B + C = 180^\circ \) or \( A + B + C = \pi \).

Using the identity for the tangent of a sum:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
we can express \( \tan C \) as:
\[ \tan C = \tan(180^\circ - (A + B)) = -\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

### Step 2: Substitute \( C \) in Terms of \( A \) and \( B \)
Now, substituting \( \tan C \) into the equation, we have:
\[ \tan A + \tan B + \tan C = \tan A + \tan B + \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

### Step 3: Simplify the Expression
Let \( x = \tan A \) and \( y = \tan B \). Then:
\[ \tan A + \tan B + \tan C = x + y + \frac{x + y}{1 - xy} \]
This can be simplified to:
\[ = x + y + \frac{x + y}{1 - xy} = (x + y) \left(1 + \frac{1}{1 - xy}\right) \]
This expression can be further analyzed to find its minimum value.

### Step 4: Use AM-GM Inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\tan A + \tan B + \tan C}{3} \geq \sqrt[3]{\tan A \tan B \tan C} \]
We also know that in an acute triangle:
\[ \tan A \tan B \tan C = \tan A + \tan B + \tan C \]
Thus, we can conclude:
\[ \tan A + \tan B + \tan C \geq 3\sqrt[3]{\tan A \tan B \tan C} \]

### Step 5: Find the Minimum Value
The minimum value occurs when \( A = B = C = 60^\circ \):
\[ \tan 60^\circ = \sqrt{3} \]
Thus:
\[ \tan A + \tan B + \tan C = 3\sqrt{3} \]
This confirms that the minimum value of \( \tan A + \tan B + \tan C \) is \( 3\sqrt{3} \).

### Step 6: Analyze Statement 2
The second statement is the AM-GM inequality:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
This is true for any positive real numbers \( a, b, c \).

### Conclusion
Both statements are true, and Statement 2 provides a correct explanation for Statement 1.

### Final Answer
Both Statement 1 and Statement 2 are true, and Statement 2 is a correct explanation of Statement 1.

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