Is it possible to simplify a trigonometric expression only in terms of $\cos (u)$ OR $\sin (u)$?

Question

I want to expand the following expression into powers of $\cos(u)$ only:

a1 + a2 Cos[2 u] + a3 Cos[4 u] + a4 Cos[6 u]

The answer that I want (and which I found by hand) is :

(a1 + a3 - a2 - a4) + (2 a2 - 8 a3 + 18 a4) Cos[u]^2 + (8 a3 - 48 a4) Cos[u]^4 + (32 a4) Cos[u]^6

As you see it contains only powers of $\cos (u)$. But when I use the TrigExpand, which is supposed to simplify the expression into powers of trigonometric functions, it gives this:

1 + a2 Cos[u]^2 + a3 Cos[u]^4 + a4 Cos[u]^6 - a2 Sin[u]^2 - 6 a3 Cos[u]^2 Sin[u]^2 
- 15 a4 Cos[u]^4 Sin[u]^2 + a3 Sin[u]^4 + 15 a4 Cos[u]^2 Sin[u]^4 - a4 Sin[u]^6

Is it possible to make TrigExpand to simplify only in terms of $\cos (u)$ or $\sin (u)$? (or maybe using another command instead of TrigExpand)

By the way, It is my first work done in Mathematica.

Answer 1

expr = a1 + a2 Cos[2 u] + a3 Cos[4 u] + a4 Cos[6 u];

(expr // TrigExpand) /. Sin[x_] -> (1 - Cos[x]^2)^(1/2) // Collect[#, Cos[_]] &

a1 - a2 + a3 - a4 + (2 a2 - 8 a3 + 18 a4) Cos[ u]^2 + (8 a3 - 48 a4) Cos[u]^4 + 32 a4 Cos[u]^6

(expr // TrigExpand) /. Cos[x_] -> (1 - Sin[x]^2)^(1/2) // Collect[#, Sin[_]] &

a1 + a2 + a3 + a4 + (-2 a2 - 8 a3 - 18 a4) Sin[ u]^2 + (8 a3 + 48 a4) Sin[u]^4 - 32 a4 Sin[u]^6

Answer 2

Some alternative approaches can be found in How to expand tan(x+y) as normal form. Here is my answer from that link, applied to your expression:

expr = a1 + a2 Cos[2 u] + a3 Cos[4 u] + a4 Cos[6 u];

Simplify[TrigExpand[expr /. u -> ArcSin[a]]] /. a -> Sin[u]

(*
==> a1 + a2 + a3 + a4 - 2 a2 Sin[u]^2 - 8 a3 Sin[u]^2 - 
 18 a4 Sin[u]^2 + 8 a3 Sin[u]^4 + 48 a4 Sin[u]^4 - 32 a4 Sin[u]^6
*)

You can also add

Collect[%, Sin[u]]
(*
==> a1 + a2 + a3 + a4 + (-2 a2 - 8 a3 - 18 a4) Sin[
   u]^2 + (8 a3 + 48 a4) Sin[u]^4 - 32 a4 Sin[u]^6
*)