A ray of light from a denser medium strikes a rarer medium at angle of incidence i. The reflected and refracted rays make an angle of 90° with each other. The angles of reflection and refraction are r & r' respectively. The critical angle is

$\sin^{-1} (\tan r)$

Applying Snell’s Law for refraction,

$\frac{\sin i}{\sin r}=\frac{n_2}{n_1}$   …(1)

From the given condition, r + r' = 90

⇒ Sin r' = Cos r …(2)

Solution of (1) and (2) yields,

$\frac{\sin i}{\cos i}=\frac{n_2}{n_1}$  …(3)

According to the Law for refraction;

i = r …(4)

Using (3) and (4) we obtain

$⇒\tan i=\frac{n_2}{n_1}$   ….(5)

Also,

$\sin θ_c=\frac{n_2}{n_1}$, Using (5) we obtain $θ_c=\sin^{-1}(\tan i)=\sin^{-1}(\tan r)$