# cos cos &#x2061;<!-- ⁡ --> (

cos$\mathrm{cos}\left(A\pi \right)$ where $A$ is an irrational algebraic number
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Pranav Greer
Let $A$ be algebraic irrational. We claim that $\mathrm{cos}\left(A\pi \right)$ transcendental. Assume not: $\mathrm{cos}\left(A\pi \right)$ is algebraic. Then $\mathrm{sin}\left(A\pi \right)=±\sqrt{1-{\mathrm{cos}}^{2}\left(A\pi \right)}$ is algebraic. Then ${e}^{iA\pi }=\mathrm{cos}\left(A\pi \right)+i\mathrm{sin}\left(A\pi \right)$ is algebraic. Also ${e}^{iA\pi }\ne 0$ and ${e}^{iA\pi }\ne 1$. Now $2/A$ is algebraic irrational. Note
$\left({e}^{iA\pi }{\right)}^{\left(2/A\right)}={e}^{2\pi i}=1$