Suppose that two points on the unit sphere centered at the origin at $ \mathbb {R} ^ 3 $ have spherical coordinates $ (1, \: \phi_1, \: \theta_1) $, $ (1, \: \phi_2, \: \theta_2) $. Show that the dot product of the points is given by: $ \cos (\phi_2 - \phi_1) - \sin \phi_2 \sin \phi_1 (1- \cos (\theta_2 - \theta_1)) $.
Advance: Satisfy
$$\sqrt{\rho^2\sin^2\phi_1 \cos^2\theta_1 + \rho^2\sin^2\phi_1 \sin^2\theta_1 + \rho^2\cos^2 \phi_1}=1$$ $$\Rightarrow \sqrt{\sin^2\phi_1 \cos^2\theta_1 + \sin^2\phi_1 \sin^2\theta_1 + \cos^2 \phi_1}=1,$$ $$\sqrt{\rho^2\sin^2\phi_2 \cos^2\theta_2 + \rho^2\sin^2\phi_2 \sin^2\theta_2 + \rho^2\cos^2 \phi_2}=1$$ $$\Rightarrow \sqrt{\sin^2\phi_2 \cos^2\theta_2 + \sin^2\phi_2 \sin^2\theta_2 + \cos^2 \phi_2}=1$$
because $\rho = 1$.
I think this could be used as your norm, and applying the formula for the dot product of two vectors
$$\vec{u} \cdot \vec{v}= ||u||\: ||v||\: \cos \alpha $$
we have what we want. Well that's my idea.
We will first convert the vectors to cartesian coordinates and name them $u_1,u_2$ as $ (\sin(\phi_1)\cos(\theta_1),\sin(\phi_1)\sin(\theta_1),\cos(\phi_1)),(\sin(\phi_2)\cos(\theta_2),\sin(\phi_2)\sin(\theta_2),\cos(\phi_2)) $
Clearly their dot product is $$ \cos(\phi_1)\cos(\phi_2) + \sin(\phi_1)\sin(\phi_2)\left( \sin(\theta_1)\sin(\theta_2) + \cos(\theta_1)\cos(\theta_2) \right) $$
And I think now you can continue