The minimum value of `tan B tan C` in an acute angled triangle ABC is

Options

A. `tan"" ( A)/( 2)`
B. `cot "" ( A)/( 2)`
C. `cosec^(2) "" ( A)/( 2)`
D. `cot^(2)""(A)/(2)`

Answer

To find the minimum value of \( \tan B \tan C \) in an acute-angled triangle \( ABC \), we can use the following steps:

### Step 1: Use the identity for \( \tan(B + C) \)
In triangle \( ABC \), we know that \( B + C = 90^\circ \) (since \( A + B + C = 180^\circ \)). Therefore, we can express \( \tan(B + C) \) as:
\[ \tan(B + C) = \tan(90^\circ) \text{ is undefined, but we can use the identity: } \]
\[ \tan(B + C) = \frac{\tan B + \tan C}{1 - \tan B \tan C} \]
Since \( B + C = 90^\circ \), we can also say:
\[ \tan(90^\circ) \text{ approaches infinity, implying } 1 - \tan B \tan C = 0 \Rightarrow \tan B \tan C = 1 \]

### Step 2: Set \( x = \tan B \tan C \)
Let \( x = \tan B \tan C \). From the identity derived, we can rearrange it:
\[ 1 - x = 0 \Rightarrow x = 1 \]

### Step 3: Analyze the minimum value
To find the minimum value of \( \tan B \tan C \), we can use the AM-GM inequality:
\[ \tan B + \tan C \geq 2\sqrt{\tan B \tan C} \]
Let \( y = \tan B \tan C \). Then:
\[ \tan B + \tan C \geq 2\sqrt{y} \]
Since \( \tan B + \tan C \) must be positive in an acute triangle, we can derive that:
\[ y \geq 1 \]

### Step 4: Conclusion
Thus, the minimum value of \( \tan B \tan C \) occurs when \( \tan B = \tan C = 1 \) (which corresponds to \( B = C = 45^\circ \)), leading to:
\[ \tan B \tan C = 1 \]

### Final Answer
The minimum value of \( \tan B \tan C \) in an acute-angled triangle \( ABC \) is \( \boxed{1} \).

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