Let $x=a \tan \theta \Rightarrow \frac{x}{a}=\tan \theta$ $\Rightarrow \theta=\tan ^{-1}\left(\frac{x}{a}\right)$
$\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right)$
$=\tan ^{-1}\left(\frac{3 a^{2} \cdot a \tan \theta-a^{3} \tan ^{3} \theta}{a^{3}-3 a \cdot a^{2} \tan ^{2} \theta}\right)$
$=\tan ^{-1}\left(\frac{3 a^{3} \tan \theta-a^{3} \tan ^{3} \theta}{a^{3}-3 a^{3} \tan ^{2} \theta}\right)$
$=\tan ^{-1}(\tan 3 \theta)$
$=3 \theta$
$=3 \tan ^{-1} \frac{x}{a}$
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