A ray of light from a denser medium strikes a rarer medium at an angle of incidence i. If the reflected and refracted rays are mutually perpendicular to each other then what is the value of critical angle?

The situation in accordance with the given problem is shown in figure.
Applying Snell's law at the boundary at C,
$\Rightarrow \mu =\frac{{\mu }_D}{{\mu }_R}=\frac{\sin r^{\text{'}}}{\sin i}\mathrm{....}\left(i\right)$

But according to given problem

$r^{\text{'}}+{90}^0+r={180}^0$

$r^{\text{'}}+r={90}^0$      i.e.,  $r^{\text{'}}={90}^0-r$

(or)

$r^{\text{'}}={90}^0-i={180}^0\left[as\angle r=\angle i\right]$

So equation (i) reduces to 

$\mu =\frac{\sin \left({90}^0-i\right)}{\sin i}=\frac{\cos i}{\sin i}=\cot i$….. (ii)

But by definition,  ${\theta }_c={\sin }^{-1}\frac{1}{\mu }$

and from equation (ii) $\mu =\cot i$

so,  ${\theta }_c={\sin }^{-1}\left[\frac{1}{\cot i}\right]={\sin }^{-1}\left(\tan i\right)$