prism

(a) Let the refracted angle at P be angle a. Hence, by Snell's Law,
sin(theta)/sin(a) = n --------------- (1)
where n is the refractive index of the prism

Since the light ray is refracted at Q by 90 degrees, the incident angle is the critical angle c. Thus, sin(c) = 1/n

But from geometry, a + c = 90
or c = 90 - a
i.e. sin(c) = sin(90 - a)
1/n = sin(90 - a)
or 1/n = cos(a) ------------- (2)

Using the trigonometrical relation: [sin(a)]^2 + [cos(a)]^2 = 1
but from (1): sin(a) = sin(theta)/n
and from(2): cos(a) = 1/n
hence, [sin(theta)/n]^2 + (1/n)^2 = 1
i.e. n^2 = 1 + [sin(theta)]^2
n = square-root[1 + [sin(theta)]^2]

(b) (i) when angle (theta) increases slightly, angle a also increases. Since (a + c) =90, the increase of a leads to a decrease of c. The angle incident at Q is now smaller than the critical angle. The light ray will emerge out from the prism.

(ii) Similarly, when angle (theta) decreases slightly, angle a decreases. Thus the incident angle at Q increases. This makes the incident angle greater than the critical angle. The light ray will undergo total internal reflection.