We are given the equation:

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

We need to find \( \sin^4 x - \cos^4 x \).

### Step 1: Simplify the given equation

Start by solving for \( \frac{\sin x + \cos x}{\sin x - \cos x} = 3 \):

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

Expanding the right-hand side:

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

Now, bring all the \( \sin x \) terms to one side and \( \cos x \) terms to the other side:

\[
\sin x - 3\sin x = -3\cos x - \cos x
\]

\[
-2\sin x = -4\cos x
\]

Simplifying:

\[
\sin x = 2\cos x
\]

### Step 2: Use \( \sin x = 2\cos x \)

We can now use the identity \( \sin^2 x + \cos^2 x = 1 \) to find the values of \( \sin x \) and \( \cos x \).

Substitute \( \sin x = 2\cos x \) into the identity:

\[
(2\cos x)^2 + \cos^2 x = 1
\]

\[
4\cos^2 x + \cos^2 x = 1
\]

\[
5\cos^2 x = 1
\]

\[
\cos^2 x = \frac{1}{5}
\]

Taking the square root of both sides:

\[
\cos x = \frac{1}{\sqrt{5}}
\]

Now, using \( \sin x = 2\cos x \):

\[
\sin x = 2 \times \frac{1}{\sqrt{5}} = \frac{2}{\sqrt{5}}
\]

### Step 3: Find \( \sin^4 x - \cos^4 x \)

We know that:

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

Since \( \sin^2 x + \cos^2 x = 1 \), the expression simplifies to:

\[
\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x
\]

Now, calculate \( \sin^2 x - \cos^2 x \):

\[
\sin^2 x = \left( \frac{2}{\sqrt{5}} \right)^2 = \frac{4}{5}
\]

\[
\cos^2 x = \frac{1}{5}
\]

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

Thus, the value of \( \sin^4 x - \cos^4 x \) is:

\[
\boxed{\frac{3}{5}}
\]

---

Would you like any details or further clarification on any steps?

Here are 5 follow-up questions you might be interested in:

1. How can we apply this method to other trigonometric identities?
2. Can you derive a general solution for equations of this type?
3. How does the Pythagorean identity help in solving trigonometric equations?
4. What are the possible ranges for \( \sin x \) and \( \cos x \) based on this result?
5. Can we solve this equation using alternative methods, like graphing?

**Tip**: Always remember to simplify trigonometric identities using known identities like \( \sin^2 x + \cos^2 x = 1 \) to help reduce complexity.

sinx + cosx / sinx - cosx = 3, find sin^4x - cos^4x

Learn how to solve the trigonometric equation sinx + cosx / sinx - cosx = 3 and find sin^4x - cos^4x. This step-by-step guide covers key trigonometric identities and solving techniques, suitable for students in Grades 10-12.

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