In a `!ABC` , 2r = `r_(1)` and A=30 , then cos `(B - C)/2` is equal to

Options

A. `(3sqrt3)/(2sqrt2)`
B. `(3(sqrt3-1))/(2sqrt2)`
C. `(3(sqrt3-1))/(2sqrt3)`
D. none of these

Explanation

To solve the problem, we need to find the value of \( \cos \left( \frac{B - C}{2} \right) \) given that \( 2r = r_1 \) and \( A = 30^\circ \).

### Step-by-Step Solution:

1. **Understanding the relationship between r and r1**:
- We know that \( r \) is the inradius and \( r_1 \) is the exradius opposite to angle A. The relationship given is \( 2r = r_1 \).

2. **Using the formula for inradius and exradius**:
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