If `A+B+C=90^(@)," then "(cot A+ cot B+ cot C)/(cot A cot B cot C)` is equal to

Options

A. 1
B. cot A cos B cot C
C. `-1`
D. 0

Answer

To solve the problem, we need to find the value of \((\cot A + \cot B + \cot C) / (\cot A \cot B \cot C)\) given that \(A + B + C = 90^\circ\).

### Step-by-Step Solution:

1. **Use the Given Information**:
Since \(A + B + C = 90^\circ\), we can express \(A + B\) as:
\[ A + B = 90^\circ - C \]

2. **Apply Cotangent Identity**:
We know that:
\[ \cot(A + B) = \cot(90^\circ - C) = \tan C \]
Using the cotangent addition formula:
\[ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} \]
We can set this equal to \(\tan C\):
\[ \frac{\cot A \cot B - 1}{\cot A + \cot B} = \tan C \]

3. **Rewrite \(\tan C\)**:
Since \(\tan C = \frac{1}{\cot C}\), we can rewrite the equation as:
\[ \frac{\cot A \cot B - 1}{\cot A + \cot B} = \frac{1}{\cot C} \]

4. **Cross Multiply**:
Cross multiplying gives us:
\[ (\cot A \cot B - 1) \cot C = \cot A + \cot B \]

5. **Rearranging the Equation**:
Rearranging the equation, we have:
\[ \cot A \cot B \cot C - \cot C = \cot A + \cot B \]
Which can be rewritten as:
\[ \cot A + \cot B + \cot C = \cot A \cot B \cot C \]

6. **Final Expression**:
Now we can express \(\frac{\cot A + \cot B + \cot C}{\cot A \cot B \cot C}\):
\[ \frac{\cot A + \cot B + \cot C}{\cot A \cot B \cot C} = 1 \]

### Conclusion:
Thus, the value of \(\frac{\cot A + \cot B + \cot C}{\cot A \cot B \cot C}\) is equal to \(1\).