$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$?
$2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates?
Thank you very much for the help and discussion ahead.
The way to calculate the shortest distance between two points is to join the points by curves which are called "minimal geodesics". To find a minimal geodesic, you find a great circle which contains both the points. This is done by forming a circular segment between the centre of the sphere and the two points. Then the standard formula applies: $$l = r \theta = \theta$$
To compute $\theta$, we use the dot product rule $$ \cos \theta = \frac{a \cdot b}{\|a\| \|b\|} = a \cdot b$$ knowing that both $\|a\| = \|b\| = 1$. Therefore
$$l = \arccos(a \cdot b)$$
To answer the second part of your question, suppose the curve you are given is defined by a curve $r : [0,1] \rightarrow \mathbb{R}^3$. The standard formula for arclength of a curve will give you the curves length:
$$ l = \int_0^1 ||r'(t)|| dt $$
Suppose now $s(t) = (1, s_{\theta}(t), s_{\phi}(t))$ is instead presented in spherical coordinates. We then wish to know the corresponding arc length when converted into cartesian coordinates. Simply define $$r(t) := (\sin (s_{\theta}(t)) \cos(s_{\phi}(t)), \sin (s_{\theta}(t)) \sin( s_{\phi}(t)), cos(s_{\theta}(t))) $$ and use the above formula.