A rectangular glass slab ABCD, of-refractive index ${\mathrm{n}}_1$, is immersed in water of refractive index ${\mathrm{n}}_2. \left({\mathrm{n}}_1>{\mathrm{n}}_2\right)$. A ray of light in incident at the surface AB of the slab as shown.
The maximum value of the angle of incidence ${\mathrm{\alpha }}_{\max }$, such that the ray comes out only from the other surface CD is given by

Ray comes out from CD, means rays after refraction from AB get, total internally reflected at AD

$\begin{array}{l}\frac{{\mathrm{n}}_1}{{\mathrm{n}}_2}=\frac{{\mathrm{sin\alpha }}_{\max }}{\sin {\mathrm{r}}_1}\Rightarrow {\mathrm{\alpha }}_{\max }={\sin }^{-1}\left[\frac{{\mathrm{n}}_1}{{\mathrm{n}}_2}{\mathrm{sinr}}_1\right]....\left(\mathrm{i}\right)\\ \mathrm{Also}{\mathrm{r}}_1+{\mathrm{r}}_2={90}^{\mathrm{o}}\Rightarrow {\mathrm{r}}_1=90-{\mathrm{r}}_2=90-\mathrm{C}\\ {\mathrm{r}}_1=90-{\sin }^{-1}\left(\frac{1}{{ }_1{\mathrm{\mu }}_2}\right)\Rightarrow {\mathrm{r}}_1=90-{\sin }^{-1}\left(\frac{{\mathrm{n}}_2}{{\mathrm{n}}_1}\right)....\left(\mathrm{ii}\right)\\ \mathrm{hence}\mathrm{from}\mathrm{equations}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)\\ {\mathrm{\alpha }}_{\max }={\sin }^{-1}\left[\frac{{\mathrm{n}}_1}{{\mathrm{n}}_2}\sin \left\{90-{\sin }^{-1}\frac{{\mathrm{n}}_2}{{\mathrm{n}}_1}\right\}\right]\\ ={\sin }^{-1}\left[\frac{{\mathrm{n}}_1}{{\mathrm{n}}_2}\cos \left({\sin }^{-1}\frac{{\mathrm{n}}_2}{{\mathrm{n}}_1}\right)\right]\end{array}$