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Let $f(x)$ be the minimal polynomial of $\sin{2^{\circ}}$ over $\mathbb Q$, and $g(x)$ be the minimal polynomial of $\sin{5^{\circ}}$ over $\mathbb Q$, then $f(x)+f(-x)=2 g(x)\tag 1$.

I find this through my computer, since $f(x)=4096 x^{12}-12288 x^{10}-512 x^9+13824 x^8+1152 x^7-7168 x^6-864 x^5+1680 x^4+248 x^3-144 x^2-24 x+1$

and $g(x)=4096 x^{12}-12288 x^{10}+13824 x^8-7168 x^6+1680 x^4-144 x^2+1$.

Can you prove $(1)$ without figure out $f(x)$ and $g(x)$?

lsr314
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  • This discussion might be useful http://math.stackexchange.com/questions/112938/when-is-sin-x-an-algebraic-number-and-when-is-it-non-algebraic – Andreas Caranti Jan 18 '14 at 15:20

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