Question

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

Answer 1

$\begin{array}{c}y={\tan }^{-1}\left(\frac{3a^{2}x-x^3}{a^3-3x^2}\right)\\ puttingx=a\tan \theta \\ \theta ={\tan }^{-1}\frac{x}{a}\\ \Rightarrow y={\tan }^{-1}\left[\frac{3a^2\tan \theta -a^3{\tan }^3\theta }{a^3-3a^3{\tan }^2\theta }\right]\\ \Rightarrow y={\tan }^{-1}\left[\frac{3\tan \theta -{\tan }^3\theta }{1-3{\tan }^2\theta }\right]\\ \Rightarrow y={\tan }^{-1}\left[\tan 3\theta \right]\\ \Rightarrow y=3\theta \\ \Rightarrow y=3{\tan }^{-1}\frac{x}{a}\end{array}$