Solve: sin(2tan⁻¹x) = 1

Let $\tan^{-1}x = \theta$, so $x = \tan\theta$. $\sin 2\theta = 1$, so $2\theta = \frac{\pi}{2}$ and $\theta = \frac{\pi}{4}$. $x = \tan\frac{\pi}{4} = 1$. Check: $\sin(2\tan^{-1}1) = \sin\frac{\pi}{2} = 1$. Hence $x = 1$.