Suppose
![\tan \alpha = \dfrac{p}{q}](/media/m/1/e/e/1ee4c97c81a27b1925707b2b9b2ccd1a.png)
, where
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
are integers and
![q \neq 0](/media/m/6/1/a/61a9c86b8cd03eedb3200cdbda676fc4.png)
. Prove that the number
![\tan \beta](/media/m/a/9/5/a952a2055fb80b004194a00793aebdc4.png)
for which
![\tan {2 \beta} = \tan {3 \alpha}](/media/m/f/0/d/f0ddbff15d146809210641da2bd229aa.png)
is rational only when
![p^2 + q^2](/media/m/1/5/3/153910b7bf3c67fc348f1bc6d34b0a40.png)
is the square of an integer.
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Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.