Crazy Cosine Carousel

Algebra Level 5

We define a carousel to be a (unordered) set of four pairwise distinct real numbers {t1,t2,t3,t4},\{t_1,t_2,t_3,t_4\}, all strictly between 00 and 2π,2\pi , such that in some order they satisfy the following system of equations

{cos(2t1)=4cost1cost2cos(2t2)=4cost2cost3cos(2t3)=4cost3cost4cos(2t4)=4cost4cost1. \begin{cases} \cos (2t_1) = 4\cos t_1 \cos t_2\\ \cos (2t_2) = 4\cos t_2 \cos t_3\\ \cos (2t_3) = 4\cos t_3 \cos t_4\\ \cos (2t_4) = 4\cos t_4 \cos t_1. \end{cases}

How many carousels are there?

Details and assumptions

A set is unordered. If a set satisfies the above system for several different orderings, it is only counted once.

A set of values is called pairwise distinct if no two of them are equal. For example, the set {1,2,2} \{1, 2, 2\} is not pairwise distinct, because the last two values are the same.

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