The angle of the prism is A and B if the angle of minimum deviation is `(180^(@) - 2A)` then the refractive index of the material of the prism is

Options

A. `cos((A)/(2))`
B. `sin ((A)/(2))`
C. `tan ((A)/(2))`
D. `cot ((A)/(2))`

Answer

To find the refractive index of the material of the prism given the angle of the prism \( A \) and the angle of minimum deviation \( D = 180^\circ - 2A \), we can use the formula for the refractive index \( n \) of a prism:

\[ n = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step-by-Step Solution:

1. **Identify the given values**:
- Angle of the prism \( A \)
- Angle of minimum deviation \( D = 180^\circ - 2A \)

2. **Calculate \( A + D \)**:
\[ A + D = A + (180^\circ - 2A) = 180^\circ - A \]

3. **Calculate \( \frac{A + D}{2} \)**:
\[ \frac{A + D}{2} = \frac{180^\circ - A}{2} = 90^\circ - \frac{A}{2} \]

4. **Substitute into the refractive index formula**:
\[ n = \frac{\sin\left(90^\circ - \frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

5. **Use the co-function identity**:
\[ \sin(90^\circ - x) = \cos(x) \]
Therefore,
\[ n = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

6. **Simplify using the cotangent function**:
\[ n = \cot\left(\frac{A}{2}\right) \]

### Final Result:
The refractive index of the material of the prism is:
\[ n = \cot\left(\frac{A}{2}\right) \]