If `(sintheta+costheta)/(sintheta-costheta)=3`, then find the value of `sin^(4)theta-cos^(4)theta`.

Options

A. a) `(1)/(5)`
B. b) `(2)/(5)`
C. c) `(3)/(5)`
D. d) `(4)/(5)`

Answer

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

### Step 1: Rewrite the given equation
We start with the equation:
\[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3 \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \sin \theta + \cos \theta = 3(\sin \theta - \cos \theta) \]
Expanding the right side:
\[ \sin \theta + \cos \theta = 3\sin \theta - 3\cos \theta \]

### Step 3: Rearrange the equation
Rearranging the terms leads to:
\[ \sin \theta + \cos \theta - 3\sin \theta + 3\cos \theta = 0 \]
This simplifies to:
\[ -2\sin \theta + 4\cos \theta = 0 \]
or:
\[ 2\sin \theta = 4\cos \theta \]
Dividing both sides by 2 gives:
\[ \sin \theta = 2\cos \theta \]

### Step 4: Express in terms of tangent
Dividing both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)):
\[ \tan \theta = 2 \]

### Step 5: Use a right triangle to find sine and cosine
In a right triangle where \(\tan \theta = 2\), we can denote the opposite side as 2 and the adjacent side as 1. Using the Pythagorean theorem, the hypotenuse \(h\) is:
\[ h = \sqrt{2^2 + 1^2} = \sqrt{5} \]

### Step 6: Calculate sine and cosine
Now we can find \(\sin \theta\) and \(\cos \theta\):
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2}{\sqrt{5}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{5}} \]

### Step 7: Calculate \(\sin^4 \theta - \cos^4 \theta\)
We know that:
\[ \sin^4 \theta - \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^2 \theta - \cos^2 \theta) \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\), we need to find \(\sin^2 \theta - \cos^2 \theta\):
\[ \sin^2 \theta = \left(\frac{2}{\sqrt{5}}\right)^2 = \frac{4}{5}, \quad \cos^2 \theta = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5} \]
Thus:
\[ \sin^2 \theta - \cos^2 \theta = \frac{4}{5} - \frac{1}{5} = \frac{3}{5} \]

### Step 8: Final calculation
Now substituting back:
\[ \sin^4 \theta - \cos^4 \theta = 1 \cdot \frac{3}{5} = \frac{3}{5} \]

### Final Answer
The value of \(\sin^4 \theta - \cos^4 \theta\) is:
\[ \frac{3}{5} \]