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$\tan(n\theta)$ is rational

Source: BStat-BMath 2017: problem 1

May 14, 2017
algebra

Problem Statement

Let the sequence {an}n≥1\{a_n\}_{n\ge 1} be defined by an=tan⁡(nθ)a_n=\tan(n\theta) where tan⁡θ=2\tan\theta =2. Show that for all nn, ana_n is a rational number which can be written with an odd denominator.
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