Solution
= Solution
Use the <geometric series> $(1+u)^{-1}=1-u+u^2+O(u^3)$ with $u=\sin x=x+O(x^3)$. The constant term counts as the first nonzero term, so the requested expansion is
$$
\boxed{\frac1{1+\sin x}=1-x+x^2+O(x^3).}
$$
= Solution
Use the <geometric series> $(1+u)^{-1}=1-u+u^2+O(u^3)$ with $u=\sin x=x+O(x^3)$. The constant term counts as the first nonzero term, so the requested expansion is
$$
\boxed{\frac1{1+\sin x}=1-x+x^2+O(x^3).}
$$