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.

  1, 3, 5, 7

Step-by-step explanation:

You can use the identity ...

Then ...

So, {a, b, c, d} = {1, 3, 5, 7}.

Overkill method: consider the expansion of :

Expand the right hand side and compare the real parts:

Rewrite the sines in terms of cosines by using the angle sum identity.

Then using the other variant of the same identity,

so that .

bfgbh bhf gfdzf bhgfrdfgbn cvdgsytrghvn ncvhgfdtrseawa hgdrsxcvbgfrewq

step-by-step explanation:

-3.5 is the distance.

i've had this question so choose the one in the absolute value sign.