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

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. `(1)/(5)`

Answer

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

### Step 1: Use the difference of squares
We can rewrite \( \cos^4 \theta - \sin^4 \theta \) using the difference of squares formula:
\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta + \sin^2 \theta)(\cos^2 \theta - \sin^2 \theta) \]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we have:
\[ \cos^4 \theta - \sin^4 \theta = 1 \cdot (\cos^2 \theta - \sin^2 \theta) = \cos^2 \theta - \sin^2 \theta \]
Thus, we can rewrite the equation as:
\[ \cos^2 \theta - \sin^2 \theta = \frac{1}{3} \]

### Step 2: Substitute \( \sin^2 \theta \)
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can substitute:
\[ \cos^2 \theta - (1 - \cos^2 \theta) = \frac{1}{3} \]
This simplifies to:
\[ \cos^2 \theta - 1 + \cos^2 \theta = \frac{1}{3} \]
\[ 2\cos^2 \theta - 1 = \frac{1}{3} \]

### Step 3: Solve for \( \cos^2 \theta \)
Now, we can isolate \( \cos^2 \theta \):
\[ 2\cos^2 \theta = \frac{1}{3} + 1 \]
\[ 2\cos^2 \theta = \frac{1}{3} + \frac{3}{3} = \frac{4}{3} \]
\[ \cos^2 \theta = \frac{4}{6} = \frac{2}{3} \]

### Step 4: Find \( \sin^2 \theta \)
Now we can find \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 5: Calculate \( \tan^2 \theta \)
Now we can find \( \tan^2 \theta \):
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \]

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