find a prescription for D^2f+n^2f

I would like to have some help with the bolded part:

Let n be a positive integer and let E_n be the set of mappings f:R->R
that are given by a prescription of the form

f(x)=a₀+Σ from k=1 to n (a_kcos(kx)+b_ksin(kx))

where a_k, b_k∈R for each k.
Prove that E_k is a subspace of Map(R,R).
If f∈E_n is the zero mapping, prove that all the coefficients a_k, b_k must be 0.
[Hint. Proceed by induction. For this, find a prescription for D²f+n²f.]
Deduce that the 2n+1 functions
x->1, x->cos(kx), x->sin(kx) (k=1,...,n)
form a basis for E_n.

Thanks in advance for any help you are able to provide.