If `A+B=90^@`, then : `tanA.tanB=...`

Options

A. not defined
B. 1
C. 0
D. none of these

Answer

To solve the problem, we start with the given condition \( A + B = 90^\circ \). We need to find the value of \( \tan A \cdot \tan B \).

### Step-by-step Solution:

1. **Understanding the Angle Sum**:
Since \( A + B = 90^\circ \), we can express this in terms of the tangent function. We know that:
\[ \tan(A + B) = \tan(90^\circ) \]

2. **Using the Tangent Addition Formula**:
The tangent addition formula states:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Substituting \( A + B = 90^\circ \):
\[ \tan(90^\circ) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

3. **Recognizing the Value of \( \tan(90^\circ) \)**:
The value of \( \tan(90^\circ) \) is undefined or can be considered as infinity. This implies that the denominator of our fraction must equal zero for the fraction to approach infinity:
\[ 1 - \tan A \tan B = 0 \]

4. **Setting the Denominator to Zero**:
From the equation \( 1 - \tan A \tan B = 0 \), we can solve for \( \tan A \tan B \):
\[ \tan A \tan B = 1 \]

5. **Final Result**:
Therefore, we conclude that:
\[ \tan A \cdot \tan B = 1 \]

### Final Answer:
\[ \tan A \cdot \tan B = 1 \]