Find the number of values of x satisfying the equation: $$ \arctan\left(x-\frac{x^3}{4}+\frac{x^5}{16}-\frac{x^7}{64}\pm\cdots\right)+\operatorname{arccot}\left(x+\frac{x^2}{2}+\frac{x^3}{4}+\frac{x^4}{8}+\cdots\right)=\frac{\pi}{2} $$
I tried solving it by finding the sum of geometric progressions and then manipulating the equation, finally getting: $$ \operatorname{arccot}\left(\frac{2x}{2-x}\right)=\operatorname{arccot}\left(\frac{4x}{4+x^2}\right) $$ and then getting $\mathbf{x=0,-2}$. While solving, common ratios of those GPs had their own constraints. So now: $|\frac{-x^2}{4}|$ and $|\frac{x}{2}|<1$. But in my teacher's solution, he put an additional constraint that $0<|x|<2$.
Now $x ≠ -2$, that is logical. However, the correct answer says that there are no solutions to this equation, I fail to understand why $x=0$ can't satisfy the equation? As $\arctan(0)+\operatorname{arccot}(0)=\pi/2$
Any help would be appreciated.