For a given transparent medium with respect to vacuum, the...

According to Brewster's law, the refractive index $\mu$ of the medium is related to the polarizing angle $i_p$ by:
$\mu = \tan(i_p)$

The critical angle $i_c$ for total internal reflection for the same medium with respect to vacuum is given by:
$\sin(i_c) = \dfrac{1}{\mu}$

Substituting the value of $\mu$ from the first equation into the second, we get:
$\sin(i_c) = \dfrac{1}{\tan(i_p)}$
$\Rightarrow \sin(i_c) = \cot(i_p)$

Answer: $\sin(i_c) = \cot(i_p)$