After solving this problem, I'm convinced that the letter [math]t[/math] here is not a coincidence. The final answer, in fact, is
[math]t=\pm\,i\tan\left(\frac{k\pi}{n}\right),\quad k=0,1,2,\ldots,\lfloor\frac{n-1}{2}\rfloor.[/math]
This equation, interestingly, has [math]n[/math] solutions only when [math]n[/math]...
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