If the refractive index of the material of a prism is ((A)/(2)) ,...

The formula to calculate the refractive index $\left(\mu\right)$ of the material of the prism is given by

$\mu = \dfrac{\sin \left(\dfrac{A + {\delta}_{m}}{2}\right)}{\sin \left(\dfrac{A}{2}\right)} . . . \left(1\right)$

From equation (1), it follows that

$cot\dfrac{A}{2} = \dfrac{\sin \left(\dfrac{A + {\delta}_{m}}{2}\right)}{sin\dfrac{A}{2}}$

$⇒ cos\dfrac{A}{2} = \sin \left(\dfrac{A + {\delta}_{m}}{2}\right) ⇒ \sin \left(\dfrac{\pi}{2} - \dfrac{A}{2}\right) = \sin \left(\dfrac{A + {\delta}_{m}}{2}\right)$

$⇒ \dfrac{A + {\delta}_{m}}{2} = \dfrac{\pi}{2} - \dfrac{A}{2}$

$⇒ {\delta}_{m} = \pi - 2A$