For special orthogonal group $SO(3)$,
Define a maximal torus of $SO(3)$ as $K$, by
$$K:=\{ \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} : 0 \leq x \lt 2\pi \}$$.
Now we consider double coset $K/SO(3)/K$, by $$K/SO(3)/K:= \{KgK: g \in SO(3) \}$$ where $ KgK:=\{k_1 g k_2: k_1, k_2 \in K \}$.
Book explains that $K/SO(3)/K$ is isomorphic to colatitude $\phi \in [0. \pi]$ of Sphere $S^2$. It gives a map $KgK \to S^2$ by $k_1 g k_2 \mapsto k_1 g n=k_1 x$ for some $x \in S^2$ with north pole $n=(0, 0, 1)$.
However, I cannot show these two spaces are isomorphic in rigorous way. Could anybody explain this isomorphism?