Solution to Expansion of arctanx Problem

[azatkgz]

Problem

expansion
$$arctanx=\frac{\pi}{2}-\frac{1}{x}+o(\frac{1}{x})$$

Attempt:
arctanx=y
tany=x
for $$y=\frac{\pi}{2}-z$$ $$tan(\frac{\pi}{2}-z)=\frac{1}{tanz}=x$$
$$\frac{cosz}{sinz}=\frac{1+o(z^2)}{z+o(z^3)}=\frac{1}{z}(1+o(z^2))=x$$*
$$arctanx=y=\frac{\pi}{2}-z=\frac{\pi}{2}-\frac{1}{x}+o(\frac{1}{x})$$
But how we can convert $$\frac{1}{z}(1+o(z^2))=x$$
to $$z=\frac{1}{x}+o(\frac{1}{x})$$

 

[azatkgz]

i think I've found the answer.
$$z=\frac{1}{x}+o(\frac{z^2}{x})$$
$$arctanx=y=\frac{\pi}{2}-z$$
$$z=\frac{\pi}{2}-arctanx=o(1)(x\rightarrow\infty)$$
so
$$z=\frac{1}{x}+o(\frac{1}{x})$$