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 .

5x+10 is the correct one

hope this

x = 2

step-by-step explanation:

if sr parallel to ba then the sides bc and ac are in proportion, that is

= ← substitute values

= ( cross- multiply )

27x = 6(4x + 1)

27x = 24x + 6 ( subtract 24x from both sides )

3x = 6 ( divide both sides by 3 )

x = 2