Path connected component

Question

I have the following Problem:

Let G be a topological group. We denote by G_0 the path connected component of the identity and by G'_0 the connected component of the identity

1. Show that G_0 is a normal subgroup and that two elements of G are in the same path-connected component if and only if they are in the same G_0 coset
2. same thing but now with G'_0

So i think I know how to do the proof that they are normal subgroups, but i'm strugling with

"that two elements of G are in the same path-connected component if and only if they are in the same G_0 coset"

I also know this definition: "If H is a subgroup of G, two elements g_1 and g_2of G lie in the same (left) coset if g_1 = h*g_2 for some h in H."

I don't know how to start the proof , maybe someone has an idea/ hint for me ? Thank you very much !