If `sin^(2)theta-cos^(2)theta=(1)/(4)` , then the value of `(sin^(4)theta-cos^(4)theta)` is :

Options

A. `(3)/(4)`
B. `(1)/(4)`
C. `(1)/(16)`
D. `(1)/(2)`

Answer

To solve the equation \( \sin^2 \theta - \cos^2 \theta = \frac{1}{4} \) and find the value of \( \sin^4 \theta - \cos^4 \theta \), we can use the difference of squares identity.

### Step 1: Use the identity for the difference of squares
We know that:
\[ \sin^4 \theta - \cos^4 \theta = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta) \]

### Step 2: Simplify using the Pythagorean identity
From the Pythagorean identity, we have:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can substitute this into our equation:
\[ \sin^4 \theta - \cos^4 \theta = (\sin^2 \theta - \cos^2 \theta)(1) \]
This simplifies to:
\[ \sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta \]

### Step 3: Substitute the given value
From the problem, we know:
\[ \sin^2 \theta - \cos^2 \theta = \frac{1}{4} \]
So we can substitute this value into our equation:
\[ \sin^4 \theta - \cos^4 \theta = \frac{1}{4} \]

### Conclusion
Thus, the value of \( \sin^4 \theta - \cos^4 \theta \) is:
\[ \boxed{\frac{1}{4}} \]