See if this is what you want to do:

r1 = Subscript[C, 1] /. 
  Simplify[Solve[E^5 + E^-5 == Subscript[C, 1] + Subscript[C, 2], 
     Subscript[C, 1]]][[1]]

r2 = C1 /. Simplify[Solve[e^5 + e^(-5) == C1 + C2, C1]][[1]]

-r1 - r2

I could do it in just one equation but I chose to do it in two parts so you can see the procedure. Did this help?

You can also do:

n = Simplify[
    Solve[E^5 + E^-5 == Subscript[C, 1] + Subscript[C, 2], 
     Subscript[C, 1]]][[1]];

r1 = Lookup[Association@n, Subscript[C, 1]]