Are you sure that there is a solution?
This should give your equations
eqs = Table[ Sum[Cos[j a[i]], {i, 5}] == KroneckerDelta[j, 1], {j, 1, 12, 2}]
I think, if you know the cosines you (in a certain sense) know the alphas, so i go upon the cosines
eqs1 = (Expand[ eqs /. Cos[j_ x_] :> TrigExpand[Cos[j x]] /. Sin[x_] :> Sqrt[1 - Cos[x]^2]]) /. Cos[a[j_]] :> c[j];
eqs1 // TableForm
Renamend the c[i] gives
eqs2 = eqs1 /. {c[1] -> x, c[2] -> y, c[3] -> z, c[4] -> u, c[5] -> v};
% // TableForm
There is a solution (are there others? I think yes) for the 2nd eqaution, but not for the others
eqs2 /. {x -> (3/4)^(1/2), y -> (3/4)^(1/2), v -> (3/4)^(1/2), z -> (3/4)^(1/2), u -> (3/4)^(1/2)}
And, by the way, you have 6 equations
Length /@ {eqs, eqs1, eqs2}
for 5 alphas
eqs /. Cos[x_. a[i_]] -> a[i] /. Equal[x__, y_] -> x /. Plus -> List // Union