“Trigger my trig, aren'cha fine?"

As a child you were interested in Prof. Basu's notebook and the myriad of mathematical work there. This made you fond of what they called trigonometry, a fancy way of looking at triangles. Now that you recall, theory of integers was something he would talk a lot about with his colleague, Prof. Bhattacharya. You wonder whether that knowledge might be helpful to you now ...

Suppose $0<A,B,C<\pi/2$ and $\tan{A}>6$. Let $\tan{A}$, $\tan{B}$, $\tan{C}$ be in increasing arithmetic progression, such that $6$ doesn't divide the difference of any two of the terms.

Write the number of tuples $(\tan{A},\tan{B},\tan{C})$ such that all three terms are primes and $\tan C < 1000$.