What is the simplified form of `cos^(4)theta- sin^(4) theta` in terms of `sin theta`

Options

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

Answer

To simplify the expression \( \cos^4 \theta - \sin^4 \theta \) in terms of \( \sin \theta \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the expression:
\[ \cos^4 \theta - \sin^4 \theta \]

### Step 2: Recognize the difference of squares
Notice that this expression can be factored using the difference of squares formula:
\[ a^2 - b^2 = (a + b)(a - b) \]
Here, let \( a = \cos^2 \theta \) and \( b = \sin^2 \theta \). Thus, we can rewrite the expression as:
\[ (\cos^2 \theta + \sin^2 \theta)(\cos^2 \theta - \sin^2 \theta) \]

### Step 3: Use the Pythagorean identity
From the Pythagorean identity, we know:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
So, we can substitute this into our expression:
\[ 1 \cdot (\cos^2 \theta - \sin^2 \theta) = \cos^2 \theta - \sin^2 \theta \]

### Step 4: Rewrite \( \cos^2 \theta \) in terms of \( \sin \theta \)
We need to express \( \cos^2 \theta \) in terms of \( \sin \theta \). Using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we substitute:
\[ \cos^2 \theta - \sin^2 \theta = (1 - \sin^2 \theta) - \sin^2 \theta \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ 1 - \sin^2 \theta - \sin^2 \theta = 1 - 2\sin^2 \theta \]

### Final Answer
Thus, the simplified form of \( \cos^4 \theta - \sin^4 \theta \) in terms of \( \sin \theta \) is:
\[ \boxed{1 - 2\sin^2 \theta} \]