In a `DeltaABC,"cot"A/2+"cot"B/2+"cot"(C)/(2)=`

Options

A. 0
B. `"cot"(1)/(2)"A cot"(1)/(2)"B cot"(1)/(2)C`
C. 1
D. none of these

Answer

To solve the problem, we need to find the value of \(\frac{\cot(A/2) + \cot(B/2) + \cot(C/2)}{2}\) in triangle \(ABC\).

### Step-by-Step Solution:

1. **Understanding the Angles in Triangle**:
In triangle \(ABC\), the sum of the angles is given by:
\[ A + B + C = \pi \]
Therefore, we can express the half-angles as:
\[ \frac{A}{2} + \frac{B}{2} + \frac{C}{2} = \frac{\pi}{2} \]

2. **Rearranging the Angles**:
From the equation above, we can rearrange it to express \(C/2\):
\[ \frac{A}{2} + \frac{B}{2} = \frac{\pi}{2} - \frac{C}{2} \]

3. **Applying Cotangent**:
Now, we take the cotangent of both sides:
\[ \cot\left(\frac{A}{2} + \frac{B}{2}\right) = \cot\left(\frac{\pi}{2} - \frac{C}{2}\right) \]
Using the cotangent identity, we know:
\[ \cot\left(\frac{\pi}{2} - x\right) = \tan(x) \]
Thus, we have:
\[ \cot\left(\frac{A}{2} + \frac{B}{2}\right) = \tan\left(\frac{C}{2}\right) \]

4. **Using Cotangent Addition Formula**:
The cotangent addition formula states:
\[ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} \]
Therefore, we can write:
\[ \cot\left(\frac{A}{2} + \frac{B}{2}\right) = \frac{\cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) - 1}{\cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right)} \]

5. **Equating the Two Expressions**:
From our previous steps, we have:
\[ \frac{\cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) - 1}{\cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right)} = \tan\left(\frac{C}{2}\right) \]
We know that \(\tan\left(\frac{C}{2}\right) = \frac{1}{\cot\left(\frac{C}{2}\right)}\), hence:
\[ \cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) - 1 = \tan\left(\frac{C}{2}\right) \left(\cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right)\right) \]

6. **Final Expression**:
Rearranging gives us:
\[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right) + \cot\left(\frac{C}{2}\right) = 2 \cdot \cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) \]
Therefore, we can conclude:
\[ \frac{\cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right) + \cot\left(\frac{C}{2}\right)}{2} = \cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) \]

### Final Answer:
Thus, the value of \(\frac{\cot(A/2) + \cot(B/2) + \cot(C/2)}{2}\) is equal to \(\cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right)\).