Angle a is divided into two parts A and B such that A - B = x and tan A:tan B = p:q. The value of sin x is equal to:

Options

A. A. `((p+q)sin alpha)/(p-q)`
B. B. `(p sin alpha)/(p+q)`
C. C. `(p sin alpha)/(p-q)`
D. D. `((p-q)sin alpha)/(p+q)`

Answer

To solve the problem step by step, we need to analyze the given information and apply trigonometric identities.

### Step 1: Understand the problem
We have two angles A and B such that:
- \( A - B = x \)
- The ratio of their tangents is given by \( \frac{\tan A}{\tan B} = \frac{p}{q} \)

### Step 2: Use the tangent ratio
From the ratio of tangents, we can express it in terms of sine and cosine:
\[ \frac{\tan A}{\tan B} = \frac{\sin A / \cos A}{\sin B / \cos B} = \frac{\sin A \cdot \cos B}{\sin B \cdot \cos A} = \frac{p}{q} \]

### Step 3: Apply the component and dividend method
Using the component and dividend method, we can rewrite the equation:
\[ \frac{\sin A \cos B + \cos A \sin B}{\sin A \cos B - \cos A \sin B} = \frac{p + q}{p - q} \]

### Step 4: Recognize sine addition and subtraction
The numerator can be recognized as:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
And the denominator can be recognized as:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
Thus, we can rewrite our equation as:
\[ \frac{\sin(A + B)}{\sin(A - B)} = \frac{p + q}{p - q} \]

### Step 5: Substitute known values
Since we know \( A - B = x \), we can substitute:
\[ \frac{\sin(A + B)}{\sin x} = \frac{p + q}{p - q} \]
Let \( A + B = \alpha \). Then we can rewrite this as:
\[ \sin x = \frac{(p - q) \sin \alpha}{(p + q)} \]

### Step 6: Final expression
Thus, we have:
\[ \sin x = \frac{(p - q) \sin \alpha}{(p + q)} \]

### Conclusion
The value of \( \sin x \) is given by:
\[ \sin x = \frac{(p - q) \sin \alpha}{(p + q)} \]