Here the requirement is that i>c
⇒sini>sinc⇒sini>μ2μ1 ....(i)
From Snell's law μ1=sinαsinr ....(ii)
Also in △OBA
r+i=90o⇒r=(90−i)
Hence from equation (ii)
sinα=μ1sin(90−i)
⇒cosi=sinαμ1
sini=√1−cos2i=√1−(sinαμ1)2 ....(iii)
From equation (i) and (iii) √1−(sinαμ1)2>μ2μ1
⇒sin2α<(μ21−μ22)⇒sinα<√μ21−μ22
αmax=sin−1√μ21−μ22