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In "The Tangent Analogues of the Chebyshev Polynomials" by Jack S. Calcut, the calculation implies that this is true except when n is of the form $2{{p}^{k}}$for p an odd prime and k a natural number. He shows earlier that $\tan (\pi /n)$ has degree $\varphi (n)/2\text{ or }\varphi \text{(n)}$depending on whether 4|n or not. My primary interest is in the related case of ${{\tan }^{2}}(\pi /n)$ because it always has degree $\varphi (n)/2$ so it is a natural choice as generator for the maximal real subfield of the cyclotomic field $\mathbb{Q}{{({{\zeta }_{n}})}^{{}}}$ - which is usually written $\mathbb{Q}{{({{\zeta }_{n}})}^{+}}$. The traditional generator for this subfield is ${{\lambda }_{n}}=2\cos (2\pi /n)$ - which is always an algebraic integer, but of course ${{\tan }^{2}}(\pi /n)$ is not always an algebraic integer.

mezzoctane
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sunshineghh
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    This is the same as asking when $\left[ \frac{\zeta -1}{\zeta +1} \right]$ is an integer, and Mathematica appears to agree with Calcut, but there should be a simple proof. – sunshineghh Feb 21 '18 at 07:04

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