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 find the value of \( \sec(A + B) \) in a right triangle \( ABC \) where \( C \) is the right angle, we can follow these steps:

### Step 1: Understand the triangle
In triangle \( ABC \), since it is right-angled at \( C \), we know that:
- \( \angle C = 90^\circ \)
- The sum of angles in a triangle is \( 180^\circ \).

### Step 2: Use the angle sum property
According to the angle sum property:
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting \( \angle C = 90^\circ \):
\[ \angle A + \angle B + 90^\circ = 180^\circ \]

### Step 3: Solve for \( \angle A + \angle B \)
Rearranging the equation gives:
\[ \angle A + \angle B = 180^\circ - 90^\circ = 90^\circ \]

### Step 4: Find \( \sec(A + B) \)
Now, we need to find \( \sec(A + B) \):
\[ \sec(A + B) = \sec(90^\circ) \]

### Step 5: Recall the definition of secant
The secant function is defined as:
\[ \sec(x) = \frac{1}{\cos(x)} \]
Thus,
\[ \sec(90^\circ) = \frac{1}{\cos(90^\circ)} \]

### Step 6: Evaluate \( \cos(90^\circ) \)
We know that:
\[ \cos(90^\circ) = 0 \]
So,
\[ \sec(90^\circ) = \frac{1}{0} \]

### Step 7: Conclusion
Since division by zero is undefined, we conclude that:
\[ \sec(A + B) \text{ is not defined.} \]

### Final Answer:
The value of \( \sec(A + B) \) is **not defined**.

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