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

Options

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

Explanation

To solve the problem, we need to find the angle of minimum deviation (Δ_min) for a prism with a refracting angle (A) and a refractive index (μ) given by cot(A/2).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Refracting angle of the prism: A
- Refractive index of the prism material: μ = cot(A/2)

2. **Using the Formula for Refractive Index**:
- The refractive index (μ) for a prism in terms of the angle of minimum deviation (Δ_min) is given by:
\[ \mu = \frac{\sin\left(\frac{A + \Delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Substituting the Given Refractive Index**:
- We substitute μ = cot(A/2) into the formula:
\[ \cot\left(\frac{A}{2}\right) = \frac{\sin\left(\frac{A + \Delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

4. **Expressing Cotangent in Terms of Sine and Cosine**:
- Recall that cot(x) = cos(x)/sin(x):
\[ \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin\left(\frac{A + \Delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

5. **Cross Multiplying**:
- Cross-multiplying gives:
\[ \cos\left(\frac{A}{2}\right) \cdot \sin\left(\frac{A}{2}\right) = \sin\left(\frac{A + \Delta_{min}}{2}\right) \]

6. **Using the Identity for Sine**:
- We can express cos(A/2) as sin(90° - A/2):
\[ \sin\left(90^\circ - \frac{A}{2}\right) = \sin\left(\frac{A + \Delta_{min}}{2}\right) \]

7. **Setting Up the Equation**:
- From the sine identity, we have:
\[ 90^\circ - \frac{A}{2} = \frac{A + \Delta_{min}}{2} \]

8. **Solving for Δ_min**:
- Rearranging gives:
\[ 180^\circ - A = A + \Delta_{min} \]
- Therefore:
\[ \Delta_{min} = 180^\circ - 2A \]

9. **Final Answer**:
- The angle of minimum deviation is:
\[ \Delta_{min} = 180^\circ - 2A \]