I got $$F(x)=64x^7-112x^5+56x^3-7x$$How do I convert this to a polynomial with leading coefficient 1?
I believe this problem is impossible because 2 different polynomials cannot be equal for all values. The exact statement of the problem as copied from the book:
"Find a polynomial $F(x)$ with leading coefficient 1 such that $F(\cos a)=\cos(7a)$ for any angle $a$."
If I'm right, I might've found a typo in my problem book.
Sure looks as if your polynomial is correct. (How likely is it that it'd work on $f(\cos(8) =\cos(56)$ without being correct?)
And your argument that if there were a monic polynomial $G$ with the same property, it'd have to equal $F$...that's pretty solid too, although it's not quite correct; what's true is that $G(x) = F(x)$ for infinitely many values of $x$ (all between $-1$ and $1$). But that's enough to prove that they're equal -- you don't need to know, a priori, that they also agree on values outside that interval.
So it looks to me as if your problem book has an error.