If `cos^(4)theta-sin^(4)theta=(2)/(3)`, then the value of `1-2sin^(2)theta` is

Options

A. `(4)/(3)`
B. 0
C. `(2)/(3)`
D. `(1)/(3)`

Answer

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

\[ \cos^4 \theta - \sin^4 \theta = \frac{2}{3} \]

### Step 1: Apply the difference of squares formula

We can use the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\). Here, we can rewrite \(\cos^4 \theta\) and \(\sin^4 \theta\) as:

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

### Step 2: Use the Pythagorean identity

From the Pythagorean identity, we know that:

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

Thus, substituting this into our equation gives:

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

### Step 3: Set up the equation

Now, we can set this equal to the right side of the original equation:

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

### Step 4: Express \(\cos^2 \theta\) in terms of \(\sin^2 \theta\)

We know that \(\cos^2 \theta = 1 - \sin^2 \theta\). Substituting this into the equation gives:

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

### Step 5: Simplify the equation

This simplifies to:

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

### Step 6: Solve for \(1 - 2\sin^2 \theta\)

We can see that the left side of the equation is exactly what we are looking for:

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

Thus, the value of \(1 - 2\sin^2 \theta\) is:

\[ \boxed{\frac{2}{3}} \]