Derive the relation, `mu=frac(sin(A+delta_m)//2)(sin A//2)`when refraction of light takes place through a glass prism. here every letter has its usual meaning.

Explanation

`delta=i+e-A`
This figure shows the variation of angle of deviation `(delta)` with the angle of incidence (i)
When I is increased `delta` decreases, reaches a minimum value and then increases again.
For one values of `delta` thre are two angles of incidence `i_(1)` and `i_(2)`.

But at minimum deviation i.e. when `delta=delta_(m)` there isonly one angle of incidence
When the prism is placed in minimum deviation position then prism lies symmetrically w.r.t incident ray and emergent ray i.e. angle of incidence is equal to angle of emergence
`implies` When `delta=delta_(m)` then `i=e` adn `r_(1)=r_(2)` (say)
We know `r_(1)=r_(2)=A` [from eqn 4]
`r+r=A`
`2r=A`
`r=A//2`
From eq. 5
`delta=i+e-A`
`delta_(m)=i+i-A" "[ :. i=e]`
`delta_(m)+A=2i`
`i=(delta_(m)+A)/2`
Refractive index of material of prism is given by
`mu=(sini)/(sinr)`
`mu=("sin"(A+delta_(m))/2)/("sinA/2)`
which is the required prism formula.