Solve for x: 5tan⁻¹ x + 3cot⁻¹ x = 2π

$\cot^{-1}x = \frac{\pi}{2} - \tan^{-1}x$ $5\tan^{-1}x + 3\left(\frac{\pi}{2} - \tan^{-1}x\right) = 2\pi$ $2\tan^{-1}x = 2\pi - \frac{3\pi}{2} = \frac{\pi}{2}$ $\tan^{-1}x = \frac{\pi}{4}$ Hence $x = 1$.