`sin^(4) theta + cos^(4) theta` in terms of ` sin theta` is ______

Options

A. `2 sin^(4) theta -2 sin^(2) theta-1`
B. `2 sin^(4) theta-2 sin^(2)theta+1`
C. `2 sin^(4) theta + 2 sin^(2) theta-1`
D. `2 sin^(4)theta-2 sin^(2)theta`

Explanation

To express \( \sin^4 \theta + \cos^4 \theta \) in terms of \( \sin \theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin^4 \theta + \cos^4 \theta \]

### Step 2: Use the identity for sum of squares
We can use the identity \( a^2 + b^2 = (a + b)^2 - 2ab \) where \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \):
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta \]

### Step 3: Simplify using the Pythagorean identity
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sin^4 \theta + \cos^4 \theta = 1^2 - 2\sin^2 \theta \cos^2 \theta \]
\[ = 1 - 2\sin^2 \theta \cos^2 \theta \]

### Step 4: Substitute \( \cos^2 \theta \)
Now, we can express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Substituting this into the expression:
\[ = 1 - 2\sin^2 \theta (1 - \sin^2 \theta) \]

### Step 5: Expand the expression
Now we expand the expression:
\[ = 1 - 2\sin^2 \theta + 2\sin^4 \theta \]

### Step 6: Rearrange the expression
Rearranging gives us:
\[ = 2\sin^4 \theta - 2\sin^2 \theta + 1 \]

### Final Expression
Thus, the expression \( \sin^4 \theta + \cos^4 \theta \) in terms of \( \sin \theta \) is:
\[ 2\sin^4 \theta - 2\sin^2 \theta + 1 \]

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