$(a)\;\sqrt{1+\sin i} \\ (b)\;\sqrt{1+ \cos i} \\ (c)\;\sqrt{1+\cos^2 i}\\ (d)\;\sqrt{1+\sin ^2 i} $

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Applying Snell's law at top surface.

$\mu \sin r =\sin i $ -----(i)

For total internal reflection the vertical face,

$\mu \sin \theta_c=1$

Using geometry $\theta_c=90^{\circ}-r$

$\therefore \mu \sin (90-r)=1$

or $\mu \cos r =1$

On squaring and adding equation (i) and (ii) we get

$\mu^2 \sin^2 r +y^2 \cos ^2 r =1+ \sin ^2 i$

or $\mu =\sqrt {1+\sin ^2 i}$

Hence d is the correct answer.

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