There are four positive integers $a$, $b$, $c$, and $d$ such that \[ 4\cos(x)\cos(2x)\cos(4x) = \cos(ax) + \cos(bx) + \cos(cx) + \cos(dx) \]for all values of $x$. Answer with $a, b, c, d$ in any order, separated by commas.
Accepted Solution
A:
Answer: 1, 3, 5, 7Step-by-step explanation:You can use the identity ... [tex]2\cos{(a)}\cos{(b)}=\cos{(a+b)}+\cos{(a-b)}[/tex]Then ...[tex]4\cos{(x)}\cos{(2x)}\cos{(4x)}=2\cos{(x)}(\cos{(4x+2x)}+\cos{(4x-2x)})\\\\=\cos{(6x+x)}+\cos{(6x-x)}+\cos{(2x+x)}+\cos{(2x-x)}\\\\=\cos{(x)}+\cos{(3x)}+\cos{(5x)}+\cos{(7x)}[/tex]So, {a, b, c, d} = {1, 3, 5, 7}.