For an isosceles prism of angle A and refractive index `mu`, it is found that the angle of minimum deviation `delta_(m) = A`. Which of the following options is/ are correct ?

Options

A. At minimum deviation, the incident angle `i_(1)` and the refracting angle `r_(1)` at the first refracting surface are related by `r_(1) =((i_(1))/(2))`
B. For this prism, the refractive index `mu` and the angle of prism A are related as `A = (1)/(2)cos^(-1)((mu)/(2))`
C. For the angle of incidence `i_(1)` = A, the ray inside the prism is parallel to the base of the prism
D. For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is `i_(1)=sin^(-1)[sinAsqrt(4cos^(2)"(A)/(2)-1)-cosA]`

Explanation

To solve the problem, we need to analyze the situation involving an isosceles prism with an angle \( A \) and a refractive index \( \mu \). We are given that the angle of minimum deviation \( \delta_m = A \). We will derive the relationships and check which options are correct based on this information.

### Step-by-Step Solution:

1. **Understanding the Prism Configuration**:
- An isosceles prism has two equal angles at its base. Let the angle at the apex be \( A \).
- The angles at the base will each be \( \frac{180^\circ - A}{2} \).

2. **Minimum Deviation Condition**:
- At minimum deviation, the angle of incidence \( I_1 \) and the angle of emergence \( I_2 \) are equal, i.e., \( I_1 = I_2 \).
- The refracting angles at the first and second surfaces are denoted as \( R_1 \) and \( R_2 \).

3. **Relation Between Angles**:
- At minimum deviation, the relationship between the incident angle \( I_1 \) and the refracting angle \( R_1 \) is given by:
\[ R_1 = \frac{I_1}{2} \]
- Similarly, for the second surface, we have:
\[ R_2 = \frac{I_2}{2} \]

4. **Using the Triangle of Angles**:
- The sum of angles in the triangle formed by the prism gives:
\[ R_1 + R_2 + A = 180^\circ \]
- Substituting \( R_1 \) and \( R_2 \):
\[ \frac{I_1}{2} + \frac{I_1}{2} + A = 180^\circ \]
- This simplifies to:
\[ I_1 + A = 180^\circ \implies I_1 = 180^\circ - A \]

5. **Calculating Minimum Deviation**:
- The angle of minimum deviation \( \delta_m \) is given by:
\[ \delta_m = I_1 - R_1 + I_2 - R_2 \]
- Since \( I_1 = I_2 \) and \( R_1 = R_2 \):
\[ \delta_m = 2(I_1 - R_1) \]
- Substituting \( R_1 = \frac{I_1}{2} \):
\[ \delta_m = 2\left(I_1 - \frac{I_1}{2}\right) = 2\left(\frac{I_1}{2}\right) = I_1 \]
- Given \( \delta_m = A \), we have:
\[ A = I_1 \]

6. **Refractive Index Relation**:
- The relationship between the refractive index \( \mu \), angle \( A \), and minimum deviation \( \delta_m \) is given by:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
- Substituting \( \delta_m = A \):
\[ \mu = \frac{\sin\left(A\right)}{\sin\left(\frac{A}{2}\right)} \]

### Conclusion:
Based on our analysis, we can conclude that the relationships derived are consistent with the properties of the isosceles prism under the condition of minimum deviation.