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 .

answer: since the question is talking about growth you have to use the growth formula which is inital amount (1 + growth)^time

step-by-step explanation:

the intial amount is 1200, the growth is 4.8, and the time is 7. plug those numbers into the equation, also i think you have to change 4.8 into a percent. but, i have forgotton how to do that. if this is a online class go on to the lesson.