How can I derive this relation from Snell's law?

In summary, Snell's law is a mathematical equation that relates the angle of incidence and the angle of refraction when light passes through different mediums. To use it, you will need information about the refractive indices and the angles. The formula is n<sub>1</sub>sinθ<sub>1</sub> = n<sub>2</sub>sinθ<sub>2</sub>. It can be applied to light waves and other types of waves with known refractive indices. According to Snell's law, the angles of incidence and refraction are inversely proportional. It has many practical uses such as designing eyeglasses and understanding light behavior in different materials and in meteorology.
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Here, it's shown how white light, after passing from air to another medium, gets broken down into its constituent coloured rays. Each has its own refractive index in the medium, but it's only shown here red, blue and yellow. The auther comments on this image and says that, for small angles of incidence, the difference between the angles of refraction of blue and red is proportional to the difference of their respective indeces of refraction, which is trivially obtained from snell's law. I however can't derive this proportionality from Snell's law. Can someone help me?
 
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What have you tried?
 
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