To get the coefficients of `cos(x)^4`

in terms of the family `cos(k*x)`

, you can try:

```
sage: f = cos(x)^4
sage: g = f.trig_reduce() ; g
1/2*cos(2*x) + 1/8*cos(4*x) + 3/8
sage: C = [g.coefficient(cos(k*x)) for k in range(5)] ; C
[0, 0, 1/2, 0, 1/8]
```

But as you can see, Maxima gives a wrong answer since it is not able to understand that 3/8 is the coefficients in `cos(0*x)`

, so you have to recover it:

```
sage: h = sum([C[k]*cos(k*x) for k in range(len(C))]) ; h
1/2*cos(2*x) + 1/8*cos(4*x)
sage: C[0] = g-h ; C
[3/8, 0, 1/2, 0, 1/8]
```

On the way back, can build the sum and then simplify the sum as follows:

```
sage: C = [3/8, 0, 1/2, 0, 1/8]
sage: sum([C[k]*cos(k*x) for k in range(len(C))]).trig_simplify()
cos(x)^4
```

So we can go back and forth from `cos(x)^4`

to its coefficients in the family `(cos(k*x))`

automatically.

If you want to do the same for the family `(1/2 - cos(k*x))`

, this is just linear algebra, going from one basis to another. For this, you just have to build the matrix `M`

that express the family `(1/2 - cos(k*x))`

in terms of the family `(cos(k*x))`

(you should be careful of the Maxima error as well so that the coefficient 1/2 will not disapear) and apply the inverse of this matrix to C (viewed as a vector), and you get:

```
sage: D
(-11/8, 0, -1/2, 0, -1/8)
sage: sum([D[k]*(1/2-cos(k*x)) for k in range(len(D))]).trig_simplify()
cos(x)^4
```