To express [tex]\(\cos 3x\)[/tex] in terms of [tex]\(\cos x\)[/tex], we need to arrange the given steps in the correct logical order. Here is the detailed, step-by-step process:
1. Start with the angle sum formula for cosine:
[tex]\[
\cos(3x) = \cos(2x + x)
\][/tex]
2. Apply the angle addition formula for cosine:
[tex]\[
\cos(3x) = \cos(2x) \cos(x) - \sin(2x) \sin(x)
\][/tex]
3. Express [tex]\(\cos(2x)\)[/tex] and [tex]\(\sin(2x)\)[/tex] in terms of [tex]\(\cos(x)\)[/tex] and [tex]\(\sin(x)\)[/tex]:
[tex]\[
\cos(2x) = 1 - 2\sin^2(x) \quad \text{and} \quad \sin(2x) = 2 \sin(x) \cos(x)
\][/tex]
Substitute these into the formula:
[tex]\[
\cos(3x) = (1 - 2\sin^2(x)) \cos(x) - (2 \sin(x) \cos(x)) \sin(x)
\][/tex]
4. Simplify the expression:
[tex]\[
\cos(3x) = [1 - 2\sin^2(x)] \cos(x) - [2 \sin(x) \cos(x)] \sin(x)
\][/tex]
[tex]\[
\cos(3x) = \cos(x) - 2\sin^2(x) \cos(x) - 2\sin(x) \cos(x) \sin(x)
\][/tex]
[tex]\[
\cos(3x) = \cos(x) - 4\sin^2(x) \cos(x)
\][/tex]
5. Rewrite [tex]\(\sin^2(x)\)[/tex] as [tex]\(1 - \cos^2(x)\)[/tex]:
[tex]\[
\cos(3x) = \cos(x) - 4(1 - \cos^2(x)) \cos(x)
\][/tex]
[tex]\[
\cos(3x) = \cos(x) - 4[\sin^2(x)] \cos(x)
\][/tex]
[tex]\[
\cos(3x) = \cos(x)\left\{1 - 4[1 - \cos^2(x)]\right\}
\][/tex]
6. Simplify further:
[tex]\[
\cos(3x) = \cos(x)\left[1 - 4 + 4\cos^2(x)\right]
\][/tex]
[tex]\[
\cos(3x) = \cos(x)\left[-3 + 4\cos^2(x)\right]
\][/tex]
7. Final form:
[tex]\[
\cos(3x) = 4\cos^3(x) - 3\cos(x)
\][/tex]
Therefore, the correct order of steps to express [tex]\(\cos 3x\)[/tex] in terms of [tex]\(\cos x\)[/tex] is:
1. [tex]\(\cos(2x + x)\)[/tex]
2. [tex]\(\cos(2x) \cos(x) - \sin(2x) \sin(x)\)[/tex]
3. [tex]\([1-2 \sin^2(x)] \cos(x) - [2 \sin(x) \cos(x)] \sin(x)\)[/tex]
4. [tex]\(\cos(x) - 4 \sin^2(x) \cos(x)\)[/tex]
5. [tex]\(\cos(x)[1 - 4 \sin^2(x)]\)[/tex]
6. [tex]\(\cos(x)\{1 - 4 [1 - \cos^2(x)]\}\)[/tex]
7. [tex]\(\cos(x)[-3 + 4 \cos^2(x)]\)[/tex]
8. [tex]\(4 \cos^3(x) - 3 \cos(x)\)[/tex]
This completes the detailed, step-by-step solution for expressing [tex]\(\cos 3x\)[/tex] in terms of [tex]\(\cos x\)[/tex].
