To express $i(x) = \cos(4x)$ as a linear combination of $f(x) = \cos^4(x)$, $g(x) = \sin^4(x)$, and $h(x) = \sin^2(2x)$, we will use trigonometric identities. Let’s proceed step by step.

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Step 1: Rewrite $\cos(4x)$ using a double angle identity

Since the hint suggests starting with $\cos(4x) = \cos(2(2x))$, we apply the double angle identity:

$\cos(2\theta) = 2\cos^2(\theta) - 1$ Let $\theta = 2x$. Then,

$\cos(4x) = 2\cos^2(2x) - 1$

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Step 2: Express $\cos^2(2x)$ in terms of $\sin^2(2x)$

We know the Pythagorean identity:

$\cos^2(2x) = 1 - \sin^2(2x)$ Substituting this into the equation for $\cos(4x)$,

$\cos(4x) = 2(1 - \sin^2(2x)) - 1$ Expanding the expression,

$\cos(4x) = 2 - 2\sin^2(2x) - 1 = 1 - 2\sin^2(2x)$

Thus, we have:

$i(x) = 1 - 2h(x)$ since $h(x) = \sin^2(2x)$.

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Step 3: Express the constant term $1$ as a linear combination of $f(x)$ and $g(x)$

We can use the identity:

$\cos^2(x) + \sin^2(x) = 1$ Squaring both sides,

$\cos^4(x) + 2\cos^2(x)\sin^2(x) + \sin^4(x) = 1$ Notice that $\cos^4(x) = f(x)$ and $\sin^4(x) = g(x)$, so the equation becomes:

$f(x) + g(x) + 2\cos^2(x)\sin^2(x) = 1$ We can rewrite $2\cos^2(x)\sin^2(x)$ using the double angle identity $\sin(2x) = 2\sin(x)\cos(x)$:

$2\cos^2(x)\sin^2(x) = \frac{1}{2}\sin^2(2x) = \frac{1}{2}h(x)$ Therefore, the equation becomes:

$1 = f(x) + g(x) + \frac{1}{2}h(x)$

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Step 4: Rearranging to express $i(x)$ in the desired form

Now, substitute $1 = f(x) + g(x) + \frac{1}{2}h(x)$ into the expression $i(x) = 1 - 2h(x)$:

$i(x) = \left(f(x) + g(x) + \frac{1}{2}h(x)\right) - 2h(x)$ Simplifying:

$i(x) = f(x) + g(x) - \frac{3}{2}h(x)$

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Final Answer:

$i(x) = f(x) + g(x) - \frac{3}{2}h(x)$ Thus, the coefficients are:

• $a = 1$
• $b = 1$
• $c = -\frac{3}{2}$

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Would you like a more detailed explanation of any part of this solution?

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Here are 5 related questions for further practice:

1. Derive the identity for $\cos(4x)$ using the complex exponential form of cosine.
2. Verify that $\cos^4(x) + \sin^4(x) + \frac{1}{2}\sin^2(2x) = 1$ for any $x$.
3. Expand $\cos^4(x)$ and $\sin^4(x)$ in terms of multiple angle identities.
4. Find the coefficients when expressing $\sin(4x)$ as a linear combination of similar functions.
5. Use the identity $\cos^2(x) = \frac{1 + \cos(2x)}{2}$ to express $\cos^4(x)$ in another form.

Tip: Knowing how to manipulate double and triple angle identities is crucial in solving complex trigonometric problems.