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})