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$OP$ is the incidence ray, which is making the angle $i_{1}$ with normal, and $QR$ is the angle of emergence, which is represented by $i_{2}$. $A$ is the prism angle and $μ$ is the refractive index of the prism.

Now, We know that,

$A=$Prism angle, $δ=$Angle of deviation, $i_{1}=$Angle of incidence, $i_{2}=$Angle of emergent.

In the case of minimum deviation,$∠r_{1}=∠r_{2}=∠r$

$A=∠r_{1}+∠r_{2}$

SO, $A=∠r+∠r=∠2r$

$∠r=2A $

Now, again

$A+δ=i_{1}+i_{2}(∵$ In the case of minimum deviation

$i_{1}=i_{2}=i$ and $δ=δ_{m})$

So, $A+δ_{m}=i+i=2i$

Now, $i=2A+δ_{m} $

Now, from snell's rule,

$μ=sinrsini $

$μ=sin2A sin2A+δ_{m} $

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