Question

Write the function in the simplest form:
${\tan }^{-1}\left(\frac{3a^{2}x-x^3}{a^3-3{ax}^2}\right),a>0;\frac{-a}{\sqrt{3}}\le x\le \frac{a}{\sqrt{3}}$

Answer 1

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

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

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

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

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

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

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