Question

Evaluate: ${\tan }^{-1}\left(\frac{3a^{2}x-x^3}{a^3-3a{x}^2}\right)$

Answer 1

Let $x=a\tan \theta$

Then,

${\tan }^{-1}\left(\frac{3a^{2}x-x^3}{a^3-2a{x}^2}\right)$

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

$={\tan }^{-1}\left(\frac{3\tan \theta -{\tan }^3\theta }{1-2{\tan }^2\theta }\right)$

$={\tan }^{-1}\tan 3\theta$

$=3\theta =3{\tan }^{-1}\frac{x}{a}$