= Solution
Define the <smooth function>
$$
F:\mathbb R^{n+1}\longrightarrow\mathbb R,
\qquad
F(x)=\lVert x\rVert^2.
$$
At $x\in F^{-1}(1)$ its derivative is
$$
D_xF(v)=2x\mathbin\cdot v.
$$
This linear map is surjective because $x\ne0$; for example, $D_xF(x)=2$. Hence one is a <regular value>, and the <regular level set theorem> makes
$$
S^n=F^{-1}(1)
$$
an embedded codimension-one submanifold of $\mathbb R^{n+1}$. Its tangent space is the kernel of the derivative:
$$
\boxed{T_xS^n
=\ker D_xF
=\{v:x\mathbin\cdot v=0\}
=x^\perp}.
$$
This is the <unit sphere as a regular level set>.
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