We previously derived the formula:Thank you! I understood one crucial thing: we need to substitute in pie into LHS tan(5theta) so that we get theta=pie/5, which is one of our roots. And, since tan(pie) = 0, we can set our RHS expression = 0 too. By the same intuition, tan(2pie) is zero. Thus, overall, I understood the idea behind setting tan(5theta) = 0.
But what I don't get is why did you even think about tan(pie) = 0 in the first place -- then going on to using this to solve the problem. That is, you could have instead chosen, say, tan(pie/4) = 1, but you didn't. Why not? What gave you the idea to use tan(pie) = 0?