Question

Solve for $x$:

${\tan }^{-1}x=\frac{1}{2}{\cot }^{-1}x$.

Answer 1

Given,

${\tan }^{-1}x=\frac{1}{2}{\cot }^{-1}x$

or, $2{\tan }^{-1}x={\cot }^{-1}x$

or, ${\tan }^{-1}\frac{2x}{1-x^2}={\tan }^{-1}\frac{1}{x}$

or, $\frac{2x}{1-x^2}=\frac{1}{x}$

or, $2x^2=1-x^2$

or, $3x^2=1$

or, $x=\pm \frac{1}{\sqrt{3}}$.