\(\frac{{{n_1}}}{{{n_2}}} = \frac{{\sin {\alpha _{\max }}}}{{\sin {r_1}}} \Rightarrow {\alpha _{\max }} = {\sin ^{ - 1}}\left[ {\frac{{{n_1}}}{{{n_2}}}\sin {r_1}} \right]\) …\((i) \)

Also \({r_1} + {r_2} = {90^o} \Rightarrow {r_1} = 90 - {r_2} = 90 - C\)

\( \Rightarrow \)\({r_1} = 90 - {\sin ^{ - 1}}\left({\frac{1}{{_2{\mu _1}}}} \right) \Rightarrow {r_1} = 90 - {\sin ^{ - 1}}\left({\frac{{{n_2}}}{{{n_1}}}} \right)\) ...\((ii)\)

Hence from equation \((i)\) and \((ii)\)

\({\alpha _{\max }} = {\sin ^{ - 1}}\left[ {\frac{{{n_1}}}{{{n_2}}}\sin \left\{ {90 - {{\sin }^{ - 1}}\frac{{{n_2}}}{{{n_1}}}} \right\}} \right]\)

= \({\sin ^{ - 1}}\left[ {\frac{{{n_1}}}{{{n_2}}}\cos \left({{{\sin }^{ - 1}}\frac{{{n_2}}}{{{n_1}}}} \right)} \right]\)