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We are tasked with proving the identity:

\[
\cos(4\theta) + 4\cos(2\theta) \equiv 8\cos^4(\theta) - 3.
\]

### Step 1: Express \(\cos(4\theta)\) and \(\cos(2\theta)\) in terms of \(\cos(\theta)\).
- We use the known double-angle identities:
  \[
  \cos(2\theta) = 2\cos^2(\theta) - 1,
  \]
  \[
  \cos(4\theta) = 2\cos^2(2\theta) - 1 = 2(2\cos^2(\theta) - 1)^2 - 1.
  \]
  Let's simplify \(\cos(4\theta)\):
  \[
  \cos(4\theta) = 2(4\cos^4(\theta) - 4\cos^2(\theta) + 1) - 1.
  \]
  \[
  \cos(4\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 2 - 1 = 8\cos^4(\theta) - 8\cos^2(\theta) + 1.
  \]

### Step 2: Express the left-hand side of the identity.

We now have:
\[
\cos(4\theta) + 4\cos(2\theta) = (8\cos^4(\theta) - 8\cos^2(\theta) + 1) + 4(2\cos^2(\theta) - 1).
\]
Simplifying the terms:
\[
\cos(4\theta) + 4\cos(2\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 1 + 8\cos^2(\theta) - 4.
\]
\[
\cos(4\theta) + 4\cos(2\theta) = 8\cos^4(\theta) - 3.
\]

### Step 3: Conclusion

The identity is verified as:

\[
\cos(4\theta) + 4\cos(2\theta) = 8\cos^4(\theta) - 3.
\]

Would you like further details on any part of the proof? Here are some related questions for further exploration:

1. What are other forms of the double-angle identities?
2. How can the identity for \(\sin(4\theta)\) be derived similarly?
3. What is the general form of the multiple-angle identities?
4. How would you approach proving an identity involving \(\tan(\theta)\)?
5. Can trigonometric identities be useful in solving integrals?

**Tip:** Always check if a trigonometric identity can be simplified using standard known formulas such as double-angle, half-angle, or sum-to-product identities.

Prove the identity: cos(4θ) + 4cos(2θ) ≡ 8cos^4(θ) - 3.

Learn how to prove the trigonometric identity cos(4θ) + 4cos(2θ) ≡ 8cos^4(θ) - 3 step-by-step using double angle identities and algebraic manipulation. Suitable for high school and college-level trigonometry students.

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