$A B C$ is the prism.A ray is incident on face $A B$ .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 $(\theta_{C})$ and the angle of emergence is $90^{\circ}$ . $\;\angle APQ = 90^{\circ} - r$ $\;\angle AQP = 90^{\circ} - \theta_C$ So, in $\triangle APQ$, $\; 90^{\circ} - r + 90^{\circ} - \theta_{C} + A = 180^{\circ}$ $\therefore \; r = A - \theta_{C}$ Applying Snell's law at point $P$, $1 \times \sin \; i = \mu \times \sin \; r$ [considered the surrounding medium is air and its refractive index is 1] $\; \text{ Or, }\; \sin i = \mu \sin (A - \theta_{C})$ $\; \text{ Or, }\; \sin i = \frac{1}{\sin \theta_c} \times \sin (A - \theta_c)$ $\therefore \; i = \sin^{-1} \left[ \frac{\sin (A - \theta_c)}{\sin \theta_c} \right]$ This is the required relation.