The refracting angle of a prism is `A`, the refractive index of the material of the prism is `cot((A)/(2))`. The angle of minimum deviation is :

Options

A. `180^@ - 2 A`
B. `90^@ - A`
C. `180^@ + 2 A`
D. `180^@ - 3 A`

Explanation

To solve the problem, we will use the relationship between the refractive index of a prism, the refracting angle, and the angle of minimum deviation.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The refracting angle of the prism is denoted as \( A \).
- The refractive index \( n \) of the prism material is given as \( n = \cot\left(\frac{A}{2}\right) \).

2. **Using the Formula for Refractive Index**:
- The formula relating the refractive index \( n \), the angle of minimum deviation \( \delta_{min} \), and the refracting angle \( A \) is:
\[ n = \frac{\sin\left(\frac{A + \delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Substituting the Refractive Index**:
- We substitute \( n \) with \( \cot\left(\frac{A}{2}\right) \):
\[ \cot\left(\frac{A}{2}\right) = \frac{\sin\left(\frac{A + \delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

4. **Expressing Cotangent in Terms of Sine and Cosine**:
- Recall that \( \cot\left(\frac{A}{2}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \).
- Therefore, we can rewrite the equation as:
\[ \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin\left(\frac{A + \delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

5. **Canceling the Denominator**:
- Since \( \sin\left(\frac{A}{2}\right) \) is common in both sides (assuming it is not zero), we can cancel it:
\[ \cos\left(\frac{A}{2}\right) = \sin\left(\frac{A + \delta_{min}}{2}\right) \]

6. **Using the Co-function Identity**:
- We know that \( \cos\left(\frac{A}{2}\right) = \sin\left(90^\circ - \frac{A}{2}\right) \).
- Thus, we can equate the arguments:
\[ \frac{A + \delta_{min}}{2} = 90^\circ - \frac{A}{2} \]

7. **Solving for \( \delta_{min} \)**:
- Multiply both sides by 2:
\[ A + \delta_{min} = 180^\circ - A \]
- Rearranging gives:
\[ \delta_{min} = 180^\circ - 2A \]

### Final Answer:
The angle of minimum deviation \( \delta_{min} \) is:
\[ \delta_{min} = 180^\circ - 2A \]