You could use directly

In[25]:= FourierCosSeries[Sqrt[1 - k^2 Sin[t]^2], t, 8]
Out[25]= (2 EllipticE[k^2])/\[Pi] + (
 2 Cos[2 t] (-2 (-2 + k^2) EllipticE[k^2] + 
    4 (-1 + k^2) EllipticK[k^2]))/(3 k^2 \[Pi]) + (
 2 Cos[4 t] (-2 (16 - 16 k^2 + k^4) EllipticE[k^2] + 
    16 (2 - 3 k^2 + k^4) EllipticK[k^2]))/(15 k^4 \[Pi]) + (
 2 Cos[6 t] ((512 - 768 k^2 + 268 k^4 - 6 k^6) EllipticE[k^2] + 
    4 (-128 + 256 k^2 - 155 k^4 + 27 k^6) EllipticK[k^2]))/(
 105 k^6 \[Pi]) + (
 2 Cos[8 t] (-2 (2048 - 4096 k^2 + 2496 k^4 - 448 k^6 + 
       5 k^8) EllipticE[k^2] + 
    64 (64 - 160 k^2 + 138 k^4 - 47 k^6 + 5 k^8) EllipticK[k^2]))/(
 315 k^8 \[Pi])

In[27]:= FourierCosCoefficient[Sqrt[1 - k^2 Sin[t]^2], t, 8]
Out[27]= (2 (-2 (2048 - 4096 k^2 + 2496 k^4 - 448 k^6 + 
      5 k^8) EllipticE[k^2] + 
   64 (64 - 160 k^2 + 138 k^4 - 47 k^6 + 5 k^8) EllipticK[
     k^2]))/(315 k^8 \[Pi])

Now you have to reconstruct the denominator and the polynomials in front of EllipticE as well as in front of EllipticK in dependence from n.

Possibly you can do so by understanding how the elliptic integrals do appear in the formulae.