State generalized Snell's law of refraction through multiple parallel media. Prove the relation.
`""^(a)mu_(c)=""^(a)mu_(b)xx""^(b)mu_(c)`

Explanation

Consider a composite slab made from slab of mediums of refractive indices `mu_(b)` and `mu_(c)` and placed in a medium of refractive index `mu_(a)`
Incident a ray KL on the face AB of slab ABCD at angle i. The ray will refract along LM at angle `r_(1)`.Thus using Snell.s law of refraction.
`(sini)/(sinr_(1))=(mu_(b))/(mu_(a))=""^(a)mu_(b)`.........1

The refracted ray LM falls on common face CD at angle `r_(1)` of two slabs and refracts along MN at angle `r_(2)`. Thus using Snell.s law,
`(sinr_(1))/(sinr_(2))=(mu_(c))/(mu_(b))=""^(b)mu_(c)`...........2
The refracted ray MN fals on face EF at angle `r_(2)` and emerges along NO at angle e.Thus
`(sinr_(2))/(sine)=(mu_(a))/(mu_(c))=""^(c)mu_(a)`
Multiplying 1,2, and 3 we get
`(sini)/(sinr_(1))xx(sinr_(1))/(sinr_(2))xx(sinr_(2))/(sine)=""^(a)mu_(b)xx""^(b)mu_(c)xx""^(c)mu_(a)`
or `(sini)/(sine)=""^(a)mu_(b)xx""^(b)mu_(c)xx""^(c)mu_(a)`
Since in the case of slab incidnet ray and mergent ray are parallel to each other therefore i=e.
Hence `1=""^(a)mu_(b)xx""^(b)mu_(c)xx""^(c)mu_(a)`
or `1/(""^(c)mu_(b))=""^(a)mu_(b)xx""^(b)mu_(c)`
or `""^(a)mu_(c)=""^(a)mu_(b)xx""^(b)mu_(c)`