A ray of light from a denser medium strike a rarer medium at an angle of incidence I (see Fig). The reflected and refracted rays make as angle of 90degrees with each other. The angles of reflection and refraction are r and `r` The critical angle is

Explanation

The situation in accordance with given problem is shown in figure .

Applying Snell's law at the boundary at C
`mu _(D ) ` sin I = `mu_(R )` sin r
`rArr mu= (mu_(D))/( mu_(R )) = (sin r ')/( sin i)`
But according to given problem r' +90 + r = 180
`r'+ r = 90 i.e, r= 90 - r`
or r= 90 - i `[as /_ r = /_i]`
So equation (i ) reduces to
`mu = (sin (90- i))/( sin i ) = (cos i)/( sin i ) = cot i`
But by definition ` theta_(c ) sin^(-1) . (1)/(mu)`
and from equation (ii) `mu cot i`
So `theta = sin ^(-1) [ (1)/(cot i)] = sin^(-1) (tan i )`