Derive ${D_m} = A({n_{21}} - 1)$ for a thin prism.

(N/A) thin prism is defined as a prism with a very small prism angle $A$.
For a prism,the refractive index $n$ is given by the formula:
$n = \frac{\sin((A + D_m)/2)}{\sin(A/2)}$ ... $(1)$
Since the prism is thin,$A$ is small,and consequently,the angle of minimum deviation $D_m$ is also small.
For small angles,we can use the approximation $\sin(\theta) \approx \theta$ (in radians).
Applying this to equation $(1)$:
$n \approx \frac{(A + D_m)/2}{A/2}$
$n = \frac{A + D_m}{A}$
Multiplying both sides by $A$:
$nA = A + D_m$
$D_m = nA - A$
$D_m = A(n - 1)$
If the prism is in a medium with refractive index $n_1$ and the prism material has refractive index $n_2$,then $n = n_2/n_1 = n_{21}$.
Therefore,$D_m = A(n_{21} - 1)$.