Question #41119

This question tests the understanding of principal value branch boundaries in composite inverse trigonometric functions.

A common trap is blindly applying the identities $\tan^{-1}\left(\frac{2x}{1-x^2}\right) = 2\tan^{-1}x$ and $\tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right) = 3\tan^{-1}x$ for all $x \in \mathbb{R}$.

Students falling for this will solve $5\tan^{-1}x = \frac{\pi}{4}$ and incorrectly conclude there is only 1 real root.

The rigorous approach requires analyzing the piecewise definitions of these functions across critical points $x = \pm 1/\sqrt{3}$ and $x = \pm 1$, revealing two additional valid roots in the outer branches.