If triangle ABC is right angled at C, then the value of sec (A+B) is

Options

A. `0`
B. `1`
C. `(2)/(sqrt(3))`
D. Not defined

Explanation

To solve the problem, we need to find the value of sec(A + B) given that triangle ABC is right-angled at C.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
In triangle ABC, we know that angle C is the right angle. Therefore, angle C = 90°.

2. **Sum of Angles in a Triangle**:
The sum of the angles in any triangle is always 180°. Thus, we can write the equation:
\[ A + B + C = 180° \]
Substituting C = 90° into the equation gives:
\[ A + B + 90° = 180° \]

3. **Simplifying the Equation**:
To find A + B, we can rearrange the equation:
\[ A + B = 180° - 90° \]
This simplifies to:
\[ A + B = 90° \]

4. **Finding sec(A + B)**:
We need to find sec(A + B). Since we have determined that A + B = 90°, we substitute this into the secant function:
\[ \sec(A + B) = \sec(90°) \]

5. **Understanding secant**:
The secant function is defined as the reciprocal of the cosine function:
\[ \sec(x) = \frac{1}{\cos(x)} \]
Therefore, we can write:
\[ \sec(90°) = \frac{1}{\cos(90°)} \]

6. **Calculating cos(90°)**:
We know that:
\[ \cos(90°) = 0 \]
Thus, substituting this value gives:
\[ \sec(90°) = \frac{1}{0} \]

7. **Conclusion**:
Since division by zero is undefined, we conclude that:
\[ \sec(A + B) = \text{undefined} \]

### Final Answer:
The value of sec(A + B) is **undefined**.