Consider rational numbers $\frac{m}{n}$ and $\frac{m'}{n'}$, where $0<\frac{m}{n}, \frac{m'}{n'} <1$.

Then $$\sin^2 (\tfrac{m}{n} \pi) = 2 \sin^2 (\tfrac{m'}{n'} \pi)$$

When $\frac{m}{n} = \frac{1}{4}, \frac{3}{4}$ and $\frac{m'}{n'} = \frac{1}{6}, \frac{5}{6}$. (Both sides equal $\frac{1}{2}$.)

Is it possible that other $\frac{m}{n}$ and $\frac{m'}{n'}$ satisfy these conditions? Is it possible to prove that there are or are not other solutions?

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    $\begingroup$ Some MathJax advice: Named math operators should appear upright, and the common ones have their own MathJax code for this purpose (e.g. \sin, \log - see entry 11 in our MathJax guide). $\endgroup$ – Zev Chonoles Jul 9 '13 at 3:05

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