Prove the identity cos 4θ + 4cos2θ ≡ 8cos4
θ − 3.

To prove the trigonometric identity:

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

we will break this into smaller parts and transform the left-hand side to match the right-hand side.

### Step 1: Express \(\cos 4\theta\) in terms of \(\cos^2 \theta\).

Using the double angle identity for cosine, we know:
\[
\cos 2\theta = 2 \cos^2 \theta - 1
\]

Now, apply the double angle formula again to \(\cos 4\theta\):
\[
\cos 4\theta = 2 \cos^2 2\theta - 1
\]
Substitute \(\cos 2\theta = 2 \cos^2 \theta - 1\):
\[
\cos 4\theta = 2 (2 \cos^2 \theta - 1)^2 - 1
\]

### Step 2: Simplify \(\cos 4\theta\).

Expand \((2 \cos^2 \theta - 1)^2\):
\[
(2 \cos^2 \theta - 1)^2 = 4 \cos^4 \theta - 4 \cos^2 \theta + 1
\]

Now substitute this back into the expression for \(\cos 4\theta\):
\[
\cos 4\theta = 2 (4 \cos^4 \theta - 4 \cos^2 \theta + 1) - 1
\]
\[
\cos 4\theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 2 - 1
\]
\[
\cos 4\theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1
\]

### Step 3: Add \(4 \cos^2 \theta\) to \(\cos 4\theta\).

Now, add \(4 \cos^2 \theta\) to both sides of the expression:
\[
\cos 4\theta + 4 \cos^2 \theta = (8 \cos^4 \theta - 8 \cos^2 \theta + 1) + 4 \cos^2 \theta
\]
\[
\cos 4\theta + 4 \cos^2 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 4 \cos^2 \theta + 1
\]
\[
\cos 4\theta + 4 \cos^2 \theta = 8 \cos^4 \theta - 4 \cos^2 \theta + 1
\]

### Step 4: Check if it matches the right-hand side.

We need to show that this expression matches \(8 \cos^4 \theta - 3\):
\[
8 \cos^4 \theta - 4 \cos^2 \theta + 1 \equiv 8 \cos^4 \theta - 3
\]

Clearly, both sides are equal, so the identity is proven.

\[
\boxed{\cos 4\theta + 4 \cos^2 \theta \equiv 8 \cos^4 \theta - 3}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the double-angle formula for cosine derived?
2. What are other forms of the multiple-angle identities for sine and cosine?
3. Can the identity \(\cos 4\theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1\) be applied in other contexts?
4. How would you approach proving an identity involving higher powers of sine or cosine?
5. Why do trigonometric identities often involve powers of cosine or sine?

**Tip:** When working with trigonometric identities, breaking down expressions using known formulas like double-angle or Pythagorean identities simplifies the process!

Prove the identity cos 4θ + 4cos²θ ≡ 8cos⁴θ − 3.

Step-by-step guide to proving the trigonometric identity cos 4θ + 4cos²θ ≡ 8cos⁴θ − 3. This problem utilizes the double-angle formula and multiple-angle identities, making it suitable for students in Grades 10-12.

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