If (sinθ+cosθ)/(sinθ-cosθ)=3, determine the value of sin^4(θ)-cos^4 (θ) without the use of a calculator.

To solve for \(\sin^4(\theta) - \cos^4(\theta)\) given that \(\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3\), we can proceed as follows:

### Step 1: Express the equation in a simpler form
Given:
\[
\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3
\]
This implies:
\[
\sin \theta + \cos \theta = 3(\sin \theta - \cos \theta)
\]
Expanding the right side:
\[
\sin \theta + \cos \theta = 3\sin \theta - 3\cos \theta
\]

### Step 2: Group the like terms
Now, bring all terms involving \(\sin \theta\) to one side and all terms involving \(\cos \theta\) to the other side:
\[
\sin \theta - 3\sin \theta = -\cos \theta - 3\cos \theta
\]
This simplifies to:
\[
-2\sin \theta = -4\cos \theta
\]
Dividing both sides by \(-2\):
\[
\sin \theta = 2\cos \theta
\]

### Step 3: Express \(\sin \theta\) in terms of \(\cos \theta\)
We have:
\[
\sin \theta = 2\cos \theta
\]
Square both sides to eliminate the trigonometric function:
\[
\sin^2 \theta = 4\cos^2 \theta
\]

### Step 4: Use the Pythagorean identity
Recall the identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substituting \(\sin^2 \theta = 4\cos^2 \theta\) into this identity:
\[
4\cos^2 \theta + \cos^2 \theta = 1
\]
This simplifies to:
\[
5\cos^2 \theta = 1
\]
Thus:
\[
\cos^2 \theta = \frac{1}{5}
\]
And since \(\sin^2 \theta = 4\cos^2 \theta\):
\[
\sin^2 \theta = 4 \times \frac{1}{5} = \frac{4}{5}
\]

### Step 5: Find \(\sin^4(\theta) - \cos^4(\theta)\)
Now, use the identity:
\[
\sin^4 \theta - \cos^4 \theta = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)
\]
Since \(\sin^2 \theta + \cos^2 \theta = 1\), we only need to find \(\sin^2 \theta - \cos^2 \theta\):
\[
\sin^2 \theta - \cos^2 \theta = \frac{4}{5} - \frac{1}{5} = \frac{3}{5}
\]
Thus:
\[
\sin^4 \theta - \cos^4 \theta = \left(\frac{3}{5}\right) \times 1 = \frac{3}{5}
\]

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

Would you like further details on any of the steps? Here are some related questions:

1. How do you derive trigonometric identities from given equations?
2. Can you explain how to solve trigonometric equations involving multiple angles?
3. What is the significance of the Pythagorean identity in trigonometric problems?
4. How can we solve trigonometric equations using substitution?
5. What are some common mistakes when manipulating trigonometric identities?

**Tip:** When solving trigonometric identities, always keep the fundamental identities in mind as they can often simplify the process significantly.

Learn how to solve the expression sin^4(theta) - cos^4(theta) using trigonometric identities, specifically the Pythagorean identity. This problem involves advanced trigonometric concepts suitable for high school students studying trigonometry.

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