Prove that each equation is an identity. Cos^4(x)-Sin^4(x)=Cos2(x)

We are given the equation:

\[
\cos^4(x) - \sin^4(x) = \cos(2x)
\]

We need to prove that this is an identity, meaning that both sides are equal for all values of \(x\). 

### Step 1: Apply the Difference of Squares Formula
Notice that \(\cos^4(x) - \sin^4(x)\) is a difference of squares. We can rewrite it as:

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

### Step 2: Simplify Using a Fundamental Trigonometric Identity
We know the fundamental trigonometric identity:

\[
\cos^2(x) + \sin^2(x) = 1
\]

Substitute this into the equation:

\[
(\cos^2(x) - \sin^2(x))(1) = \cos^2(x) - \sin^2(x)
\]

### Step 3: Use the Double Angle Identity
Now, recall the double angle identity for cosine:

\[
\cos(2x) = \cos^2(x) - \sin^2(x)
\]

Thus, we have:

\[
\cos^4(x) - \sin^4(x) = \cos(2x)
\]

This shows that the left-hand side of the original equation simplifies to the right-hand side. Therefore, the equation is an identity.

### Conclusion:
\[
\cos^4(x) - \sin^4(x) = \cos(2x)
\]
is indeed an identity.

---

Would you like more details on any of the steps? Here are some related questions:

1. How can the difference of squares formula be used in other trigonometric identities?
2. What are other common double angle identities besides \(\cos(2x)\)?
3. How can we derive the double angle formula for cosine?
4. How does the Pythagorean identity \(\cos^2(x) + \sin^2(x) = 1\) help simplify trigonometric expressions?
5. How would this identity change if \(\cos^4(x) + \sin^4(x)\) were used instead?

**Tip:** When dealing with higher powers of trigonometric functions, look for algebraic identities like the difference of squares to simplify expressions.

Learn how to prove the trigonometric identity cos^4(x) - sin^4(x) = cos(2x) using key concepts like the difference of squares, the Pythagorean identity, and the double angle identity. Suitable for students in Grades 10-12.

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