Spherical coordinates on unit sphere are defined by the following transformation:
$$\begin{cases}x=\sin\theta\cos\varphi\\ y=\sin\theta\sin\varphi\\ z=\cos\theta\end{cases}$$
Are these coordinates the only possible orthogonal coordinate system on sphere, up to rotation and coordinate scaling?
Definitely not. You could also use stereographic projection from the plane, with the standard $(x,y)$ coordinates on the plane, or any other orthogonal coordinates on the plane, for that matter. Stereographic projection is conformal (see http://en.wikipedia.org/wiki/Stereographic_projection).
