Consider an $n$-sphere embedded in $n+1$ Euclidean space. What is the parameterization of that surface expressed as a Euclidean vector $\{x_1,x_2...x_n\}$. It seems like it should be
$x_1=\sin\theta_1...\sin\theta_{n-3}\sin\theta_{n-2}\sin\theta_{n-1}$
$x_2=\sin\theta_1...\sin\theta_{n-3}\sin\theta_{n-2}\cos\theta_{n-1}$
$x_3=\sin\theta_1...\sin\theta_{n-3}\cos\theta_{n-2}$
...
$x_n=\cos\theta_1$
But I have no idea how to prove this. Does anyone have any tips? My ultimate goal is to exploit it to create a general rotation operator for a quantum circuit.