Question

Solve for $x$, ${\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}$

Answer 1

${\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}$

$={\tan }^{-1}x+{\cot }^{-1}x+{\cot }^{-1}x$

$\frac{\pi }{2}+{\cot }^{-1}x=\frac{2\pi }{3}$

${\cot }^{-1}x=\frac{2\pi }{3}-\frac{\pi }{2}$

${\cot }^{-1}x=\frac{\pi }{6}$

$x=\cot \left(\frac{\pi }{6}\right)$

$=\cot \left(30\right)=\frac{\cos 30}{\sin 30}=\sqrt{3}$