How do I separate the two solutions?

Consider the differential equation
$$f”[x]==a f[x]$$
The result is a combination of trigonometric or hyperbolic functions, something like:

f=Function[{x},C[1]Cos[Sqrt[a]x]+C[2]Sin[Sqrt[a]x]

how do I separate the two functions Cos and Sin?

C[1] Cos[Sqrt[a] x] + C[2] Sin[Sqrt[a] x] /. Plus -> List
(*{C[1] Cos[Sqrt[a] x], C[2] Sin[Sqrt[a] x]}*)

sol = DSolve[f''[x] == a f[x], f, x]
f[x] /. First@sol /. C[1] -> 0
f[x] /. First@sol /. C[2] -> 0