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

Options

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

Answer

To solve the equation \( \cos^4 \theta - \sin^4 \theta = \frac{2}{3} \) and find the value of \( 2\cos^2 \theta - 1 \), we can follow these steps:

### Step 1: Use the difference of squares formula
We know that \( a^2 - b^2 = (a + b)(a - b) \). Here, we can apply this to \( \cos^4 \theta - \sin^4 \theta \):
\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta + \sin^2 \theta)(\cos^2 \theta - \sin^2 \theta) \]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we can simplify:
\[ \cos^4 \theta - \sin^4 \theta = 1 \cdot (\cos^2 \theta - \sin^2 \theta) = \cos^2 \theta - \sin^2 \theta \]

### Step 2: Set up the equation
Now we can set the equation equal to \( \frac{2}{3} \):
\[ \cos^2 \theta - \sin^2 \theta = \frac{2}{3} \]

### Step 3: Use the Pythagorean identity
We know that \( \sin^2 \theta = 1 - \cos^2 \theta \). Substituting this into our equation gives:
\[ \cos^2 \theta - (1 - \cos^2 \theta) = \frac{2}{3} \]
This simplifies to:
\[ \cos^2 \theta - 1 + \cos^2 \theta = \frac{2}{3} \]
or
\[ 2\cos^2 \theta - 1 = \frac{2}{3} \]

### Step 4: Solve for \( 2\cos^2 \theta - 1 \)
We have already isolated \( 2\cos^2 \theta - 1 \):
\[ 2\cos^2 \theta - 1 = \frac{2}{3} \]

### Final Answer
Thus, the value of \( 2\cos^2 \theta - 1 \) is:
\[ \boxed{\frac{2}{3}} \]