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

Options

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

Answer

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

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

### Step 1: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ \sin \theta + \cos \theta = 3(\sin \theta - \cos \theta) \]

### Step 2: Expand the right side
Expanding the right side, we have:

\[ \sin \theta + \cos \theta = 3\sin \theta - 3\cos \theta \]

### Step 3: Rearrange the equation
Now, rearranging the equation to bring like terms together:

\[ \sin \theta + \cos \theta - 3\sin \theta + 3\cos \theta = 0 \]

This simplifies to:

\[ -2\sin \theta + 4\cos \theta = 0 \]

### Step 4: Solve for \(\tan \theta\)
Rearranging gives:

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

Dividing both sides by \(2\cos \theta\) (assuming \(\cos \theta \neq 0\)) gives:

\[ \tan \theta = 2 \]

### Step 5: Find \(\sin^2 \theta\) and \(\cos^2 \theta\)
Using the identity \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\):

\[ \tan^2 \theta = 2^2 = 4 \implies \sin^2 \theta = 4\cos^2 \theta \]

Let \(\cos^2 \theta = x\). Then:

\[ \sin^2 \theta = 4x \]

Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\[ 4x + x = 1 \implies 5x = 1 \implies x = \frac{1}{5} \]

Thus,

\[ \cos^2 \theta = \frac{1}{5} \quad \text{and} \quad \sin^2 \theta = 4 \cdot \frac{1}{5} = \frac{4}{5} \]

### Step 6: Calculate \(\sin^4 \theta - \cos^4 \theta\)
Now, we need to find \(\sin^4 \theta - \cos^4 \theta\). We can use the difference of squares:

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

Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have:

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

### Step 7: Calculate \(\sin^2 \theta - \cos^2 \theta\)
Now, we find \(\sin^2 \theta - \cos^2 \theta\):

\[ \sin^2 \theta - \cos^2 \theta = \frac{4}{5} - \frac{1}{5} = \frac{3}{5} \]

### Final Result
Thus,

\[ \sin^4 \theta - \cos^4 \theta = \frac{3}{5} \]

The final answer is:

\[ \sin^4 \theta - \cos^4 \theta = \frac{3}{5} \]