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The equation $$\small15\arctan^2(x)-10\arctan^2(2x)-2\arctan^2(3x)+3\arctan^2(7x)=0\tag{1}$$ has real roots $\{-1, -\rho, 0, \rho, 1\},$ where $\rho\approx0.5354195932557...$

Is $\rho$ an algebraic number? Does it have any closed form?

The equation $(1)$ is interesting because it is one of few known examples of vanishing linear combinations (with non-zero integer coefficients) of squared arctangents of positive integers (see https://oeis.org/A263920).

Vladimir Reshetnikov
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    For all $\alpha\ne0,1$ algebraic $\tan(\alpha)$ is trascendental.But if $\alpha $ es trascendental, then $\tan(\alpha)$ can be trascendental or algebraic, even integer. – Ataulfo Feb 08 '25 at 01:20

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