Value of `sec^(2)theta-(sin^(2)theta-2sin^(4)theta)/(2cos^(4)theta-cos^(2)theta)` is

Options

A. 1
B. 2
C. `-1`
D. 0

Answer

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

### Step 1: Rewrite the expression using trigonometric identities
We know that \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \). Thus, we can rewrite the expression as:
\[ \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta - 2\sin^4 \theta}{2\cos^4 \theta - \cos^2 \theta} \]

### Step 2: Simplify the denominator
The denominator \( 2\cos^4 \theta - \cos^2 \theta \) can be factored:
\[ 2\cos^4 \theta - \cos^2 \theta = \cos^2 \theta (2\cos^2 \theta - 1) \]

### Step 3: Substitute back into the expression
Now, substituting the factored form back into the expression gives:
\[ \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta - 2\sin^4 \theta}{\cos^2 \theta (2\cos^2 \theta - 1)} \]

### Step 4: Combine the fractions
To combine the fractions, we need a common denominator:
\[ \frac{1 \cdot (2\cos^2 \theta - 1)}{\cos^2 \theta (2\cos^2 \theta - 1)} - \frac{\sin^2 \theta - 2\sin^4 \theta}{\cos^2 \theta (2\cos^2 \theta - 1)} \]
This simplifies to:
\[ \frac{2\cos^2 \theta - 1 - (\sin^2 \theta - 2\sin^4 \theta)}{\cos^2 \theta (2\cos^2 \theta - 1)} \]

### Step 5: Simplify the numerator
Now, simplify the numerator:
\[ 2\cos^2 \theta - 1 - \sin^2 \theta + 2\sin^4 \theta \]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can replace \( \sin^2 \theta \) with \( 1 - \cos^2 \theta \):
\[ 2\cos^2 \theta - 1 - (1 - \cos^2 \theta) + 2\sin^4 \theta \]
This further simplifies to:
\[ 3\cos^2 \theta - 2 + 2\sin^4 \theta \]

### Step 6: Substitute \( \sin^2 \theta \) in terms of \( \cos^2 \theta \)
Using \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ 2\sin^4 \theta = 2(1 - \cos^2 \theta)^2 = 2(1 - 2\cos^2 \theta + \cos^4 \theta) \]
Substituting this back gives:
\[ 3\cos^2 \theta - 2 + 2(1 - 2\cos^2 \theta + \cos^4 \theta) \]
This simplifies to:
\[ 3\cos^2 \theta - 2 + 2 - 4\cos^2 \theta + 2\cos^4 \theta = -\cos^2 \theta + 2\cos^4 \theta \]

### Step 7: Final simplification
Now we have:
\[ \frac{-\cos^2 \theta + 2\cos^4 \theta}{\cos^2 \theta (2\cos^2 \theta - 1)} \]
Factoring out \( \cos^2 \theta \) from the numerator:
\[ \frac{\cos^2 \theta(2\cos^2 \theta - 1)}{\cos^2 \theta(2\cos^2 \theta - 1)} = 1 \]

### Conclusion
Thus, the value of the expression is:
\[ \boxed{1} \]