$\mathit{A}\mathit{B}\mathit{C}$ is the prism.
A ray is incident on face $\mathit{A}\mathit{B}$ .
Angle of incidence $=\mathit{i}$
It is refracted at an angle $\mathit{r}$ and incident on AC face.
There is just suffers total internal reflection means, the angle of incidence is critical angle $\left({\mathit{\theta }}_{\mathrm{C}}\right)$ and the angle of emergence is ${90}^{\circ }$ .

$\angle \mathit{A}\mathit{P}\mathit{Q}={90}^{\circ }-\mathit{r}$
$\angle \mathit{A}\mathit{Q}\mathit{P}={90}^{\circ }-{\mathit{\theta }}_{\mathit{C}}$
So, in $△\mathrm{APQ}$,
${90}^{\circ }-\mathit{r}+{90}^{\circ }-{\mathit{\theta }}_{\mathrm{C}}+\mathit{A}={180}^{\circ }$
$\therefore \mathit{r}=\mathit{A}-{\mathit{\theta }}_{\mathrm{C}}$
Applying Snell's law at point $\mathit{P}$,
$1\times \mathit{s}\mathit{i}\mathit{n}\mathit{i}=\mathit{\mu }\times \mathit{s}\mathit{i}\mathit{n}\mathit{r}$ [considered the surrounding medium is air and its refractive index is 1]


$\therefore \mathit{i}=\mathit{s}\mathit{i}\mathit{n}{}^{-1}\left[\frac{\mathit{s}\mathit{i}\mathit{n}\left(\mathit{A}-{\mathit{\theta }}_{\mathit{c}}\right)}{\mathit{s}\mathit{i}\mathit{n}{\mathit{\theta }}_{\mathit{c}}}\right]$
This is the required relation.