If `cot A =(4)/(3) and (A+B)=90^(@)`, what is the value of tan B?

Explanation

To solve the problem, we need to find the value of \( \tan B \) given that \( \cot A = \frac{4}{3} \) and \( A + B = 90^\circ \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
Since \( A + B = 90^\circ \), we know that \( B = 90^\circ - A \). This means that \( B \) is the complementary angle to \( A \).

2. **Using the Cotangent Identity**:
We know that \( \cot A = \frac{1}{\tan A} \). Given \( \cot A = \frac{4}{3} \), we can find \( \tan A \):
\[ \tan A = \frac{1}{\cot A} = \frac{1}{\frac{4}{3}} = \frac{3}{4} \]

3. **Finding \( \tan B \)**:
Using the complementary angle identity, we have:
\[ \tan B = \cot A \]
Since \( \cot A = \frac{4}{3} \), we can directly write:
\[ \tan B = \frac{4}{3} \]

4. **Final Answer**:
Therefore, the value of \( \tan B \) is:
\[ \tan B = \frac{4}{3} \]