If `cos^2 theta - sin^2 theta = 1/3`, where `0 le theta le pi/2`, then the value of `cos^4 theta - sin^4 theta` is

Options

A. `1/3`
B. `2/3`
C. `1/9`
D. `2/9`

Answer

To solve the problem, we start with the given equation:

\[ \cos^2 \theta - \sin^2 \theta = \frac{1}{3} \]

We need to find the value of:

\[ \cos^4 \theta - \sin^4 \theta \]

### Step 1: Rewrite \(\cos^4 \theta - \sin^4 \theta\)

We can use the difference of squares formula, which states that \(A^2 - B^2 = (A - B)(A + B)\). Here, we can let:

- \(A = \cos^2 \theta\)
- \(B = \sin^2 \theta\)

Thus,

\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta) \]

### Step 2: Simplify \(\cos^2 \theta + \sin^2 \theta\)

From the Pythagorean identity, we know that:

\[ \cos^2 \theta + \sin^2 \theta = 1 \]

### Step 3: Substitute the values into the equation

Now substituting back into our expression:

\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta - \sin^2 \theta)(1) \]

This simplifies to:

\[ \cos^4 \theta - \sin^4 \theta = \cos^2 \theta - \sin^2 \theta \]

### Step 4: Substitute the value of \(\cos^2 \theta - \sin^2 \theta\)

From the original equation, we have:

\[ \cos^2 \theta - \sin^2 \theta = \frac{1}{3} \]

### Step 5: Final Calculation

Thus, we can conclude:

\[ \cos^4 \theta - \sin^4 \theta = \frac{1}{3} \]

### Final Answer

The value of \(\cos^4 \theta - \sin^4 \theta\) is:

\[ \frac{1}{3} \]

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