`sin^(4) theta + 2 cos^(2) theta (1 - (1)/( sec^(2) theta)) + cos ^(4) theta = `

Options

A. 1
B. 2
C. `sqrt(2)`
D. 0

Answer

To simplify the expression \( \sin^4 \theta + 2 \cos^2 \theta \left(1 - \frac{1}{\sec^2 \theta}\right) + \cos^4 \theta \), we will follow these steps:

### Step 1: Rewrite the expression
The given expression is:
\[ \sin^4 \theta + 2 \cos^2 \theta \left(1 - \frac{1}{\sec^2 \theta}\right) + \cos^4 \theta \]

### Step 2: Simplify the term involving secant
Recall that \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \). Thus, we can rewrite \( 1 - \frac{1}{\sec^2 \theta} \) as:
\[ 1 - \frac{1}{\sec^2 \theta} = 1 - \cos^2 \theta = \sin^2 \theta \]

### Step 3: Substitute back into the expression
Substituting this back into our expression gives:
\[ \sin^4 \theta + 2 \cos^2 \theta \sin^2 \theta + \cos^4 \theta \]

### Step 4: Recognize the form of the expression
Notice that \( \sin^4 \theta + \cos^4 \theta \) can be expressed using the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2 \sin^2 \theta \cos^2 \theta \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2 \sin^2 \theta \cos^2 \theta \]

### Step 5: Combine the terms
Now we can combine this with the \( 2 \cos^2 \theta \sin^2 \theta \) term:
\[ 1 - 2 \sin^2 \theta \cos^2 \theta + 2 \cos^2 \theta \sin^2 \theta \]
This simplifies to:
\[ 1 - 2 \sin^2 \theta \cos^2 \theta + 2 \sin^2 \theta \cos^2 \theta = 1 \]

### Final Result
Thus, the simplified expression is:
\[ \boxed{1} \]