Trigonometry equation with arctan Solve the following equation: $\arctan x + \arctan (x^2-1) = \frac{3\pi}{4}$.
What I did
Let  $\arctan x = \alpha, \arctan(x^2-1) = \beta$, $\qquad\alpha+\beta = \frac{3\pi}{4}$
$\tan(\alpha+\beta) = \tan(\frac{3\pi}{4}) = -1$
$$\frac{\tan\alpha + tan\beta}{1-\tan\alpha\tan\beta} = \frac{x+x^2-1}{1-x(x^2-1)} = -1$$
$\begin{align}
x^2+x-1 &= -(1-(x^3-x)) = -1+x^3-x \\
\iff x^2 + x &= x^3-x \\
\iff x(x+1) &= x(x^2-1) \qquad\implies \boxed{x_1 = 0}\\
\implies x+1 &= x^2-1 \\
\iff x^2-x-2 &= 0 \\
\end{align}$
$\therefore x_1 = 0,\quad x_2 = 2,\quad x_3 = -1$
However, the equation only works for $x=2$.
I wonder
Did I do this in an efficient manner? Is there any easy way to find $x$ where there's no fake solutions?

Your three answers all look good to me. You did this in a pretty efficient manner, although I would've just jumped straight to the identity $$\arctan(A)+\arctan(B) = \arctan\left(\frac{A+B}{1-AB} \right)$$ without making the substitution $\arctan(x) = \alpha$, etc. 
Anyway, you are confident that the equation works for $x=2$ (which it does.) But for $x=0$ note that a calculator will tell us that $$\arctan(0)+\arctan(0^2-1) = 0+\arctan(-1) = \frac{-\pi}{4}$$ However, this is because $$\tan\left( \frac{3\pi}{4}\right) =\tan\left(  \frac{-\pi}{4} \right)$$ where there is a discrepancy with the calculator due to the fact that Cosine is negative and Sine is positive in the quadrant where $\frac{3\pi}{4}$ lies, and Cosine is positive while Sine is negative in the quadrant where $\frac{-\pi}{4}$ lies. Long story short, the calculator doesn't know the difference in the calculation and defaults to $\frac{-\pi}{4}$. The same exact thing happens for $x = -1$. All three of your answers are right though.
