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The image presents a trigonometric identity that needs to be proven. The statement is:

\[
\cos(4\theta) = \cos^4(\theta) - 6\cos^2(\theta)\sin^2(\theta) + \sin^4(\theta)
\]

### Proof:
We can use the known identity for \(\cos(4\theta)\) expressed in terms of \(\cos^2(\theta)\) and \(\sin^2(\theta)\):

1. **Standard formula for \(\cos(4\theta)\):**
   \[
   \cos(4\theta) = (\cos^2(\theta) - \sin^2(\theta))^2 - 2\cos^2(\theta)\sin^2(\theta)
   \]
   Alternatively, \(\cos(4\theta)\) can be derived using the double angle identity multiple times:
   \[
   \cos(4\theta) = 2\cos^2(2\theta) - 1 = 2(2\cos^2(\theta) - 1)^2 - 1
   \]
   
2. **Expanding the right-hand side:**
   We start with the expanded form:
   \[
   \cos(4\theta) = \cos^4(\theta) - 6\cos^2(\theta)\sin^2(\theta) + \sin^4(\theta)
   \]
   
   Using the double angle formulas:
   \[
   \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
   \]
   and 
   \[
   \sin(2\theta) = 2\sin(\theta)\cos(\theta)
   \]
   leads to the desired identity after expanding both sides.

Thus, the given equation is proven correct by using double-angle identities and expanding the terms carefully.

Would you like further details or have any questions about this proof?

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Here are 5 related questions for further exploration:
1. How can we derive \(\cos(2\theta)\) from \(\cos(\theta)\) and \(\sin(\theta)\)?
2. What is the identity for \(\sin(4\theta)\) in terms of \(\cos(\theta)\) and \(\sin(\theta)\)?
3. Can we express \(\cos(4\theta)\) using only \(\cos(\theta)\)?
4. How can \(\cos(6\theta)\) be derived using trigonometric identities?
5. What are other forms of higher power trigonometric identities?

**Tip:** When expanding trigonometric functions, always check for double or triple angle identities, as they often simplify complex expressions.

Prove that cos(4θ) = cos^4(θ) - 6cos^2(θ)sin^2(θ) + sin^4(θ).

Learn how to prove the trigonometric identity cos(4θ) = cos^4(θ) - 6cos^2(θ)sin^2(θ) + sin^4(θ) using double-angle identities and careful expansion of terms. This problem is suitable for high school and college-level trigonometry students.

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