If `(sinA)/(sinB)=(sqrt(3))/(2) and (cosA)/(cosB)=(sqrt(5))/(2) , 0 lt A, B lt (pi)/(2)` then `tanA+tanB` is equal to :

Options

A. `sqrt((3)/(5))`
B. `sqrt((5)/(3))`
C. `(sqrt(3)+sqrt(5))/(sqrt(5))`
D. `(sqrt(3)+sqrt(5))/(sqrt(3))`

Answer

To solve the problem, we need to find the value of \( \tan A + \tan B \) given the relationships between \( \sin A \), \( \sin B \), \( \cos A \), and \( \cos B \).

### Step-by-Step Solution:

1. **Given Equations**:
\[ \frac{\sin A}{\sin B} = \frac{\sqrt{3}}{2} \quad \text{(1)} \]
\[ \frac{\cos A}{\cos B} = \frac{\sqrt{5}}{2} \quad \text{(2)} \]

2. **Divide Equation (1) by Equation (2)**:
\[ \frac{\sin A}{\sin B} \div \frac{\cos A}{\cos B} = \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{5}}{2}} \implies \frac{\tan A}{\tan B} = \frac{\sqrt{3}}{\sqrt{5}} \]
Let \( k = \frac{\tan A}{\tan B} = \frac{\sqrt{3}}{\sqrt{5}} \).

3. **Express \( \tan A \) and \( \tan B \)**:
\[ \tan A = k \tan B \implies \tan A = \frac{\sqrt{3}}{\sqrt{5}} \tan B \]

4. **Substituting \( \tan A \) in terms of \( \tan B \)**:
From the above, we have:
\[ \tan A = \frac{\sqrt{3}}{\sqrt{5}} \tan B \]

5. **Using the identity for sine and cosine**:
From Equation (1):
\[ \sin A = \frac{\sqrt{3}}{2} \sin B \implies \sin A = \frac{\sqrt{3}}{2} \cdot \frac{\tan B}{\sqrt{1 + \tan^2 B}} \]
From Equation (2):
\[ \cos A = \frac{\sqrt{5}}{2} \cos B \implies \cos A = \frac{\sqrt{5}}{2} \cdot \frac{1}{\sqrt{1 + \tan^2 B}} \]

6. **Equating the expressions**:
We can express \( \sin A \) and \( \cos A \) in terms of \( \tan B \) and solve for \( \tan B \):
\[ \frac{\sqrt{3}}{2} \cdot \frac{\tan B}{\sqrt{1 + \tan^2 B}} = \frac{\sqrt{5}}{2} \cdot \frac{1}{\sqrt{1 + \tan^2 B}} \]

7. **Cross-multiplying**:
\[ \sqrt{3} \tan B = \sqrt{5} \]

8. **Solving for \( \tan B \)**:
\[ \tan B = \frac{\sqrt{5}}{\sqrt{3}} \]

9. **Finding \( \tan A \)**:
Substitute \( \tan B \) back into the equation for \( \tan A \):
\[ \tan A = \frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{3}} = 1 \]

10. **Calculating \( \tan A + \tan B \)**:
\[ \tan A + \tan B = 1 + \frac{\sqrt{5}}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3}} + \frac{\sqrt{5}}{\sqrt{3}} = \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3}} \]

### Final Answer:
\[ \tan A + \tan B = \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3}} \]