Two angles of a triangle are `cot^-1 2` and `cot^-1 3,` then the third angle is

Options

A. `(pi)/(4)`
B. `3(pi)/(4)`
C. `(pi)/(6)`
D. `(pi)/(3)`

Answer

To find the third angle of the triangle given the first two angles as \( \cot^{-1} 2 \) and \( \cot^{-1} 3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Angles**:
Let \( A = \cot^{-1} 2 \) and \( B = \cot^{-1} 3 \). We need to find the third angle \( C \).

2. **Use the Triangle Angle Sum Property**:
The sum of the angles in a triangle is \( 180^\circ \) or \( \pi \) radians. Therefore, we have:
\[ A + B + C = \pi \]
This implies:
\[ C = \pi - (A + B) \]

3. **Combine the Angles Using Cotangent Addition Formula**:
We can use the formula for the sum of cotangents:
\[ \cot^{-1} x + \cot^{-1} y = \cot^{-1} \left( \frac{xy - 1}{x + y} \right) \]
Here, \( x = 2 \) and \( y = 3 \). Therefore:
\[ A + B = \cot^{-1} \left( \frac{2 \cdot 3 - 1}{2 + 3} \right) = \cot^{-1} \left( \frac{6 - 1}{5} \right) = \cot^{-1} \left( \frac{5}{5} \right) = \cot^{-1}(1) \]

4. **Evaluate \( A + B \)**:
Since \( \cot^{-1}(1) = \frac{\pi}{4} \), we have:
\[ A + B = \frac{\pi}{4} \]

5. **Find the Third Angle \( C \)**:
Substitute \( A + B \) back into the equation for \( C \):
\[ C = \pi - \frac{\pi}{4} = \frac{4\pi}{4} - \frac{\pi}{4} = \frac{3\pi}{4} \]

6. **Conclusion**:
Therefore, the third angle \( C \) is:
\[ C = \frac{3\pi}{4} \]

### Final Answer:
The third angle of the triangle is \( \frac{3\pi}{4} \).

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