Write the function in the simplest form:
Question:

Write the function in the simplest form:

$\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right), a>0 ; \frac{-a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}$

 

Solution:

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

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

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

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

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

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

$=3 \theta$

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

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