Certainly! Let's tackle this step-by-step.
### Part (a): Expand [tex]\(\cos(x + 2x)\)[/tex]
We start with the expression [tex]\(\cos(x + 2x)\)[/tex]. Notice that [tex]\(x + 2x\)[/tex] simplifies to [tex]\(3x\)[/tex], so we are really looking to expand [tex]\(\cos(3x)\)[/tex].
### Part (b): Express [tex]\(\cos(3x)\)[/tex] in terms of [tex]\(\cos(x)\)[/tex]
To express [tex]\(\cos(3x)\)[/tex] in terms of [tex]\(\cos(x)\)[/tex], we can use the triple angle formula for cosine. The triple angle formula states that:
[tex]\[ \cos(3x) = 4\cos^3(x) - 3\cos(x) \][/tex]
Thus, we have:
[tex]\[ \cos(3x) = 4\cos^3(x) - 3\cos(x) \][/tex]
### Final Answer
From part (a), expanding [tex]\(\cos(x + 2x)\)[/tex] we get:
[tex]\[ \cos(x + 2x) = \cos(3x) \][/tex]
And from part (b), the expanded form of [tex]\(\cos(3x)\)[/tex] in terms of [tex]\(\cos(x)\)[/tex] is:
[tex]\[ \cos(3x) = 4\cos^3(x) - 3\cos(x) \][/tex]
In summary:
- [tex]\(\cos(x + 2x) = \cos(3x)\)[/tex]
- [tex]\(\cos(3x) = 4\cos^3(x) - 3\cos(x)\)[/tex]
These steps align with the results, where [tex]\(\cos(3x)\)[/tex] is expanded and simplified to [tex]\(4\cos^3(x) - 3\cos(x)\)[/tex].
