Vectors `vecà and vecb` include an angle `theta` between them. If `(veca+vecb)` and `(veca-vecb)` respectively subtend angles `alpha and beta` with `veca`, then `(tan alpha+tan beta) ` is

Options

A. `(("ab"sin theta))/((a^(2)+b^(2)cos^(2) theta))`
B. `(2"ab"sin theta)/((a^(2)-b^(2)cos^(2) theta))`
C. `(a^(2)sin^(2) theta)/((a^(2)+b^(2)cos^(2) theta))`
D. `((b^(2)sin^(2) theta))/((a^(2)-b^(2)cos^(2) theta))`

Explanation

To solve the problem, we need to find the expression for \( \tan \alpha + \tan \beta \) where \( \alpha \) and \( \beta \) are the angles that the resultant vectors \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \) make with vector \( \vec{a} \), respectively.

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- Let \( \vec{a} \) and \( \vec{b} \) be two vectors with an angle \( \theta \) between them.
- The resultant vector \( \vec{R_1} = \vec{a} + \vec{b} \) makes an angle \( \alpha \) with \( \vec{a} \).
- The resultant vector \( \vec{R_2} = \vec{a} - \vec{b} \) makes an angle \( \beta \) with \( \vec{a} \).

2. **Finding \( \tan \alpha \)**:
- The tangent of angle \( \alpha \) can be expressed as:
\[ \tan \alpha = \frac{|\vec{b}| \sin \theta}{|\vec{a}| + |\vec{b}| \cos \theta} \]
- Here, \( |\vec{b}| \sin \theta \) is the perpendicular height from \( \vec{b} \) to the resultant \( \vec{R_1} \), and \( |\vec{a}| + |\vec{b}| \cos \theta \) is the base.

3. **Finding \( \tan \beta \)**:
- Similarly, for \( \tan \beta \):
\[ \tan \beta = \frac{|\vec{b}| \sin \theta}{|\vec{a}| - |\vec{b}| \cos \theta} \]
- Here, \( |\vec{b}| \sin \theta \) is the perpendicular height from \( -\vec{b} \) to the resultant \( \vec{R_2} \), and \( |\vec{a}| - |\vec{b}| \cos \theta \) is the base.

4. **Adding \( \tan \alpha \) and \( \tan \beta \)**:
- Now, we can add \( \tan \alpha \) and \( \tan \beta \):
\[ \tan \alpha + \tan \beta = \frac{|\vec{b}| \sin \theta}{|\vec{a}| + |\vec{b}| \cos \theta} + \frac{|\vec{b}| \sin \theta}{|\vec{a}| - |\vec{b}| \cos \theta} \]
- This can be simplified using a common denominator:
\[ = \frac{|\vec{b}| \sin \theta (|\vec{a}| - |\vec{b}| \cos \theta) + |\vec{b}| \sin \theta (|\vec{a}| + |\vec{b}| \cos \theta)}{(|\vec{a}| + |\vec{b}| \cos \theta)(|\vec{a}| - |\vec{b}| \cos \theta)} \]
- Simplifying the numerator:
\[ = \frac{2 |\vec{b}| |\vec{a}| \sin \theta}{|\vec{a}|^2 - |\vec{b}|^2 \cos^2 \theta} \]

5. **Final Result**:
- Thus, we arrive at the final expression:
\[ \tan \alpha + \tan \beta = \frac{2 |\vec{a}| |\vec{b}| \sin \theta}{|\vec{a}|^2 - |\vec{b}|^2 \cos^2 \theta} \]

### Conclusion:
The value of \( \tan \alpha + \tan \beta \) is given by:
\[ \frac{2 |\vec{a}| |\vec{b}| \sin \theta}{|\vec{a}|^2 - |\vec{b}|^2 \cos^2 \theta} \]