The refractive index of a prism with apex angle `A` is `cot A//2`. Prove that the angle of minimum deviation is `delta_m = (180^@ - 2 A)`.

Explanation

To prove that the angle of minimum deviation \( \delta_m \) for a prism with apex angle \( A \) and refractive index \( n = \cot \frac{A}{2} \) is given by \( \delta_m = 180^\circ - 2A \), we can follow these steps:

### Step 1: Write the formula for the refractive index of a prism
The refractive index \( n \) of a prism can be expressed in terms of the apex angle \( A \) and the angle of minimum deviation \( \delta_m \) as follows:

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