Fresnel Equations and Snells Law

[girlinphysics]

I have seen in many textbooks that the following Fresnel Equation $$\frac{n_t{cos(\theta_i)} - n_i{cos(\theta_t)}}{n_t{cos(\theta_i)} + n_i{cos(\theta_t)}}$$ and then they say that via Snells Law we get the following equation $$\frac{tan(\theta_i - \theta_t)}{tan(\theta_i + \theta_t)}$$

I can see that some trig manipulation was done, and I know that ${tan(\theta)} = \frac{cos(\theta)}{sin(\theta)}$ but I cannot seen to get to that final result correct. If anyone is able to show me the maths behind it that would be great.

 

[ShayanJ]

When I first encountered this, it bothered me too because I couldn't connect them. But at last I did it with a trick. I started from the second one to reach the first one, which is pretty easy. And then the proof you want is the reverse of that. Just expand the tangents and the rest is almost obvious.