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]]
