The refracting angle of a prism is $$A$$ and refractive index of the material of the prism is $$\cot\left(\frac{A}{2}\right)$$. Then the angle of minimum deviation will be

$$\mu = \cot\left(\frac{A}{2}\right)$$

$$\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$

$$\cot\left(\frac{A}{2}\right) = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$

$$\implies \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$ $$\implies \cos\left(\frac{A}{2}\right) = \sin\left(\frac{A + \delta_m}{2}\right)$$

$$\sin\left(90^\circ - \frac{A}{2}\right) = \sin\left(\frac{A + \delta_m}{2}\right)$$ $$\implies 90^\circ - \frac{A}{2} = \frac{A + \delta_m}{2}$$

$$\implies 180^\circ - A = A + \delta_m$$ $$\implies \delta_m = 180^\circ - 2A$$