Q.(a) Deduce the relation μ = sin((A+δm)/2) / sin(A/2), where the symbols have their usual meaning.

The prism formula follows from geometry of the symmetric ray path at minimum deviation; for a thin prism $\mu = 1+\delta/A$, giving $\mu=1.5$ here.

(a) Derivation: For a ray passing through a prism of refracting angle $A$, refraction occurs at both faces. Let $i_1,i_2$ be the angles of incidence and emergence, and $r_1,r_2$ the angles of refraction at the two faces. Geometry of the prism gives

$A = r_1+r_2$

and the total deviation is

$\delta = i_1+i_2-A$

At the angle of minimum deviation $\delta_m$, the ray path becomes symmetric: $i_1=i_2=i$ and $r_1=r_2=r=A/2$. Then

$\delta_m = 2i-A \;\Rightarrow\; i = \frac{A+\delta_m}{2}$

Applying Snell's law at the first face, $\mu = \dfrac{\sin i}{\sin r}$: …