The critical angle is \(\theta\) for a light going from medium \(P\) to medium Q. If speed of light in medium \(\mathrm{P}\) is \(\mathrm{V}_{\mathrm{P}}\) then speed of light in medium \(\mathrm{Q}\) is

For total internal reflection for critical incidence angle \(\theta\) the angle of refraction is \(\frac{\pi}{2}\).
Using Snell's law:
\(\mathrm{n}_{\mathrm{A}} \sin \theta=\mathrm{n}_{\mathrm{B}} \sin \left(\frac{\pi}{2}\right)\), where, \(\mathrm{n}_{\mathrm{A}}=\) refractive index of medium \(\mathrm{A}\) and \(n_B=\) refractive index of medium \(B\)
and \(\mathrm{n}_{\mathrm{B}}=\) refractive index of medium \(\mathrm{B}\)
\(\frac{n_A}{n_B}=\frac{1}{\sin \theta}\)
We define refractive index in terms of velocity of light as follows:
\(\begin{aligned} & \frac{\mathrm{n}_{\mathrm{A}}}{\mathrm{n}_{\mathrm{B}}}=\frac{\left.\mathrm{V}_{\mathrm{B}} \text { (velocity in medium } \mathrm{B}\right)}{\left.\mathrm{V}_{\mathrm{A}} \text { (velocity in medium } \mathrm{A}\right)} \\ & \therefore \frac{\mathrm{V}_{\mathrm{A}}}{\mathrm{V}_{\mathrm{B}}}=\frac{1}{\sin \theta} \\ & \mathrm{V}_{\mathrm{B}}=\frac{\mathrm{V}_{\mathrm{A}}}{\sin \theta}=\frac{\mathrm{V}}{\sin \theta}\end{aligned}\)