$(a)\;(1+\sin ^2 \theta) \\ (b)\;(\sin ^2 \theta-1)^{1/2} \\ (c)\;(1+\sin ^2 \theta)^{1/2} \\ (d)\;(\sin ^2 \theta-1)$

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Since angle of prison $A=90^{\circ}$

$\angle OPQ +\angle OQP=90^{\circ}$

Also since the emergent angle is $90^{\circ}$

$\angle OQP= \theta _c$ the critical angle

$\therefore \angle OPQ=90 - \theta _c$

Now $\mu = \large\frac{\sin \theta}{\sin (90 - \theta_l)}$

$\mu =\large\frac{\sin \theta}{\cos \theta_l}$

$\qquad= \large\frac{\sin \theta }{\sqrt {1- \sin^2 \theta_c}}$

We have $\mu =\large\frac{1}{\sin \theta_c}$

$\mu \sqrt {1- \large\frac{1}{\mu^2}}=\sin \theta$

$\mu^2 (1- \large\frac{1}{\mu^2})$$ =\sin \theta$

$\mu^2-1=\sin ^2 \theta$

$\mu^2= 1+\sin ^2 \theta$

$\mu =(1+ \sin ^2 \theta )^{1/2}$

Hence c is the correct answer

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