It is quite an elementary question but I need a little bit help to elaborate it.
Let $\text{char}\,k\neq2$. I want to show that the quotient of algebraic groups $SO_n/SO_{n-1}$ is isomorphic to the affine variety $X=\{(x_1,...,x_n)|x_1^2+...+x_n^2=1\}$. (all is over $k$.)
Take $e=(0,...,0,1)$. Then $SO_n$ acts on $X$ and the stabilizer of $e$ is equal to $SO_{n-1}\subset SO_n$. Thus we obtain a map $$\varphi:SO_n/SO_{n-1}\to X,$$ which sends $A\in SO_n$ to $Ae\in X$. The tangent space to $X$ at $e$ is given by the hyperplane $\{x_n=0\}$. Now it remains to show that $\varphi$ is separable morphism, so we I need to show that $d\varphi$ is surjective. How can I compute $d\varphi$?