This was an exercise from generatingfunctionology by H.Wilf. The question is to find inverse of power series for $\sin(x)$.
In the solution it mentions to set $x $ in $\sin(x)$ as $x+bx^2+cx^3..$, equate to $x$ and solve for $b,c...$
My question is that $f(x) = \sin(x)$ and $g(x) = f^{-1}(x) = a_0+a_1x+a_2x^2$ so why we don't replace $x$ with $a_0 + a_1x + a_2x^2..$ in $\sin(x)$ ? Doing this does give $a_0 = 0$ and $a_1 = 1$ as the book had earlier already said to apply. Now I know that $f(0) = 0 \implies g(0) = 0$ so does this mean we can omit constant term because, and also that due to $f'(0) = 1$ so graphically $y=f(x)$ is parallel to $y=x$ at origin so its mirror $g(x)$ is also same so $g'(0) = 1$ and $a_1 = 1$ ?