` (cos^(4) x - sin^(4) x)` is equal to

Options

A. `2 sin^(2) x -1`
B. `1 -2 cos^(2)x `
C. `sin^(2) x - cos^(2)x`
D. `2 cos^(2)x -1`

Explanation

To solve the expression \( \cos^4 x - \sin^4 x \), we can use the difference of squares identity. Here’s a step-by-step solution:

### Step 1: Recognize the Difference of Squares
We can express \( \cos^4 x - \sin^4 x \) as a difference of squares:
\[ \cos^4 x - \sin^4 x = (\cos^2 x)^2 - (\sin^2 x)^2 \]

### Step 2: Apply the Difference of Squares Formula
Using the difference of squares formula, \( A^2 - B^2 = (A + B)(A - B) \), we can rewrite the expression:
\[ \cos^4 x - \sin^4 x = (\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x) \]

### Step 3: Simplify the First Factor
We know from the Pythagorean identity that:
\[ \cos^2 x + \sin^2 x = 1 \]
Thus, we can substitute this into our expression:
\[ \cos^4 x - \sin^4 x = 1 \cdot (\cos^2 x - \sin^2 x) = \cos^2 x - \sin^2 x \]

### Step 4: Use the Identity for \( \cos^2 x - \sin^2 x \)
The expression \( \cos^2 x - \sin^2 x \) can be rewritten using the double angle identity:
\[ \cos^2 x - \sin^2 x = \cos(2x) \]

### Final Answer
Thus, we conclude that:
\[ \cos^4 x - \sin^4 x = \cos(2x) \]