If `Delta`ABC is right angled at C, then find the value of cos (A + B)

Answer

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

### Step-by-Step Solution:

1. **Understand the Triangle**: We know that triangle ABC is right-angled at C. This means that angle C is \( 90^\circ \).

2. **Use the Angle Sum Property**: The sum of angles in any triangle is always \( 180^\circ \). Therefore, we can write:
\[ A + B + C = 180^\circ \]

3. **Substitute the Value of C**: Since angle C is \( 90^\circ \), we can substitute this value into the equation:
\[ A + B + 90^\circ = 180^\circ \]

4. **Solve for A + B**: Rearranging the equation gives us:
\[ A + B = 180^\circ - 90^\circ = 90^\circ \]

5. **Find \( \cos(A + B) \)**: Now that we know that \( A + B = 90^\circ \), we can find \( \cos(A + B) \):
\[ \cos(A + B) = \cos(90^\circ) \]

6. **Evaluate \( \cos(90^\circ) \)**: The value of \( \cos(90^\circ) \) is:
\[ \cos(90^\circ) = 0 \]

### Final Answer:
Thus, the value of \( \cos(A + B) \) is \( 0 \).