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Let $n$ be a positive integer. Can we precisely solve the equation $\sin(x) = n\cos(x)$ in $x$?

For $n=1$, we get $x=\pi/4$.

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    By the way: 18 minutes.2011-12-01

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The only rational values of $\tan(x)$ or $\cot(x)$ for rational $x$ in degrees are $0$ and $\pm 1$. See for instance

Olmsted, J. M. H.; Discussions and Notes: Rational Values of Trigonometric Functions. Amer. Math. Monthly 52 (1945), no. 9, 507–508.

From this you cannot expect that your equation has any other nice solutions.

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    Related to this result is [Niven's theorem](htt$p$://mathworld.wolfram.com/NivensTheorem.html).2011-12-01