ABC is the prism.
A ray is incident on face AB .
Angle of incidence =i
It is refracted at an angle r and incident on AC face.
There is just suffers total internal reflection means, the angle of incidence is critical angle (θC) and the angle of emergence is 90∘ .

‌∠APQ=90∘−r
‌∠AQP=90∘−θC
So, in △APQ,
‌90∘−r+90∘−θC+A=180∘
∴‌r‌=A−θC
Applying Snell's law at point P,
1×sin‌i=µ×sin‌r [considered the surrounding medium is air and its refractive index is 1]
‌ Or, ‌‌sin‌i=µsin‌(A−θC)
‌ Or, ‌‌sin‌i=‌

1

sin‌θc

×sin‌(A−θc)
∴‌i‌=sin‌−1[‌

sin‌(A−θc)

sin‌θc

]
This is the required relation.