I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic polynomials defined on $\mathbb{R}^{n + 1}$. I was trying to derive this myself, but this is where I got stuck: I can prove that if a function is in $V_k$, then, it is the restriction of a homogeneous harmonic polynomial of degree $k$ on $\mathbb{R}^{n + 1}$, but I cannot prove the converse. That is, I cannot prove that a homogeneous harmonic polynomial of degree $k$ defined on $\mathbb{R}^{n + 1}$ restricted to $S^n$ will be an eigenfunction of $-\Delta_{S^n}$ with eigenvalue $k^2 + (n - 1)k$. Any help is appreciated, thanks.
Addendum: I see that if I can prove that a homogeneous harmonic polynomial of degree $k$ on $\mathbb{R}^{n + 1}$ can be expressed as $r^k \varphi (\omega)$, where $\omega \in S^n$, the rest follows. I am not sure how to claim this.
For completess, here is how the rest goes. The radial part of the Euclidean Laplacian (without minus sign) in $\mathbb{R}^{n+1}$ is $r^{-n}(r^n u_r)_r$ (derived here). If $u$ is a homogeneous polynomial of degree $k$, then $u(r\xi)=r^k u(\xi)$ (with $\xi\in S^n$), hence $$r^{-n}(r^n u_r)_r =- r^{-n}(k r^{n+k-1})' u(\xi) = k(n+k-1)r^{k-2}u(\xi)$$ Since $u$ is harmonic, it follows that $ -\Delta_{S^n}u = k(n+k-1) u $ as claimed.