Which of the following relations is true for two unit vector `hatA` and `hatB` making an angle `theta` to each other?

Options

A. `|hatA+hatB|=|hatA-hatB|tantheta/2`
B. `|hatA-hatB|=|hatA+hatB|tantheta/2`
C. `|hatA+hatB|=|hatA-hatB|costheta/2`
D. `|hatA-hatB|=|hatA+hatB|costheta/2`

Explanation

To solve the problem, we need to find the relationship between two unit vectors \(\hat{A}\) and \(\hat{B}\) that make an angle \(\theta\) with each other.

### Step-by-Step Solution:

1. **Understanding Unit Vectors**:
Since \(\hat{A}\) and \(\hat{B}\) are unit vectors, their magnitudes are:
\[ |\hat{A}| = 1 \quad \text{and} \quad |\hat{B}| = 1 \]

2. **Finding Magnitude of \(\hat{A} + \hat{B}\)**:
The magnitude of the sum of two vectors can be calculated using the formula:
\[ |\hat{A} + \hat{B}| = \sqrt{|\hat{A}|^2 + |\hat{B}|^2 + 2 |\hat{A}| |\hat{B}| \cos \theta} \]
Substituting the magnitudes:
\[ |\hat{A} + \hat{B}| = \sqrt{1^2 + 1^2 + 2 \cdot 1 \cdot 1 \cdot \cos \theta} = \sqrt{2 + 2 \cos \theta} = \sqrt{2(1 + \cos \theta)} = 2 \cos\left(\frac{\theta}{2}\right) \]

3. **Finding Magnitude of \(\hat{A} - \hat{B}\)**:
Similarly, for the difference of the vectors:
\[ |\hat{A} - \hat{B}| = \sqrt{|\hat{A}|^2 + |\hat{B}|^2 - 2 |\hat{A}| |\hat{B}| \cos \theta} \]
Substituting the magnitudes:
\[ |\hat{A} - \hat{B}| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos \theta} = \sqrt{2 - 2 \cos \theta} = \sqrt{2(1 - \cos \theta)} = 2 \sin\left(\frac{\theta}{2}\right) \]

4. **Finding the Relation**:
Now, we can establish a relationship between the magnitudes of the sum and difference of the vectors:
\[ \frac{|\hat{A} - \hat{B}|}{|\hat{A} + \hat{B}|} = \frac{2 \sin\left(\frac{\theta}{2}\right)}{2 \cos\left(\frac{\theta}{2}\right)} = \tan\left(\frac{\theta}{2}\right) \]

5. **Final Result**:
Therefore, the relation we have derived is:
\[ |\hat{A} - \hat{B}| = |\hat{A} + \hat{B}| \tan\left(\frac{\theta}{2}\right) \]

### Conclusion:
The correct relation for the two unit vectors \(\hat{A}\) and \(\hat{B}\) making an angle \(\theta\) with each other is:
\[ |\hat{A} - \hat{B}| = |\hat{A} + \hat{B}| \tan\left(\frac{\theta}{2}\right) \]