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

Options

A. `1+2sin^(2)""(theta)/(2)`
B. `2cos^(2)theta-1`
C. `1-2sin^(2)""(theta)/(2)`
D. `1+2cos^(2)theta`

Answer

To solve the expression \( \cos^4 \theta - \sin^4 \theta \), we can follow these steps:

### Step 1: Recognize the expression as a difference of squares
The expression \( \cos^4 \theta - \sin^4 \theta \) can be rewritten as:
\[ (\cos^2 \theta)^2 - (\sin^2 \theta)^2 \]
This is a difference of squares, which can be factored using the identity \( a^2 - b^2 = (a + b)(a - b) \).

### Step 2: Apply the difference of squares formula
Using the difference of squares formula, we factor the expression:
\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta + \sin^2 \theta)(\cos^2 \theta - \sin^2 \theta) \]

### Step 3: Use the Pythagorean identity
From the Pythagorean identity, we know that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
Thus, we can replace \( \cos^2 \theta + \sin^2 \theta \) in our expression:
\[ (\cos^2 \theta + \sin^2 \theta)(\cos^2 \theta - \sin^2 \theta) = 1 \cdot (\cos^2 \theta - \sin^2 \theta) \]

### Step 4: Simplify the expression
Now, we have:
\[ \cos^4 \theta - \sin^4 \theta = \cos^2 \theta - \sin^2 \theta \]

### Step 5: Use the double angle identity
The expression \( \cos^2 \theta - \sin^2 \theta \) can be expressed using the double angle identity for cosine:
\[ \cos^2 \theta - \sin^2 \theta = \cos(2\theta) \]

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