The angle of minimum diviation of a prism is `(pi-2A)` . Where A represents angle of the prism. The refractive index of material prism is

Options

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

Explanation

To find the refractive index (μ) of the material of a prism given that the angle of minimum deviation (D) is \( \pi - 2A \), where \( A \) is the angle of the prism, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
The angle of minimum deviation \( D \) is related to the angle of the prism \( A \) and the refractive index \( \mu \) by the formula:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

2. **Substitute the Given Values**:
From the question, we know:
\[ D = \pi - 2A \]
Substitute \( D \) into the formula:
\[ \mu = \frac{\sin\left(\frac{A + (\pi - 2A)}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Simplify the Expression**:
Simplifying the expression inside the sine function:
\[ A + (\pi - 2A) = \pi - A \]
Thus, we have:
\[ \mu = \frac{\sin\left(\frac{\pi - A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

4. **Use the Sine Identity**:
We can use the sine identity \( \sin\left(\frac{\pi}{2} - x\right) = \cos(x) \):
\[ \sin\left(\frac{\pi - A}{2}\right) = \cos\left(\frac{A}{2}\right) \]
Therefore, the equation for \( \mu \) becomes:
\[ \mu = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

5. **Convert to Cotangent**:
The ratio of cosine to sine is the cotangent function:
\[ \mu = \cot\left(\frac{A}{2}\right) \]

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