How does "casting without paying mana cost" work with additional costs?
You must pay additional costs separately, so you would still have to pay {1} for Sphere of Resistance.

Casting without paying is an alternate cost:


  117.9. Some spells have alternative costs. An alternative cost is a cost listed in a spell's text, or applied to it from another effect, that its controller may pay rather than paying the spell's mana cost. Alternative costs are usually phrased, "You may [action] rather than pay [this object's] mana cost," or "You may cast [this object] without paying its mana cost." Note that some alternative costs are listed in keywords; see rule 702.


Additional costs are added separately from alternate costs:


  117.9d If an alternative cost is being paid to cast a spell, any additional costs, cost increases, and cost reductions that affect that spell are applied to that alternative cost. (See rule 601.2f.)