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

We are given the equation \((\sin\theta + \cos\theta)(\sin\theta - \cos\theta) = 3\). Let's start by simplifying this expression.

### Step 1: Simplify the Given Expression
The given expression is a difference of squares:
\[
(\sin\theta + \cos\theta)(\sin\theta - \cos\theta) = \sin^2\theta - \cos^2\theta
\]
So, we have:
\[
\sin^2\theta - \cos^2\theta = 3
\]

### Step 2: Express \(\sin^4\theta - \cos^4\theta\) in Terms of \(\sin^2\theta - \cos^2\theta\)
Notice that \(\sin^4\theta - \cos^4\theta\) can also be expressed as a difference of squares:
\[
\sin^4\theta - \cos^4\theta = (\sin^2\theta)^2 - (\cos^2\theta)^2 = (\sin^2\theta - \cos^2\theta)(\sin^2\theta + \cos^2\theta)
\]
We know from the Pythagorean identity that:
\[
\sin^2\theta + \cos^2\theta = 1
\]
Therefore:
\[
\sin^4\theta - \cos^4\theta = (\sin^2\theta - \cos^2\theta)(1) = \sin^2\theta - \cos^2\theta
\]

### Step 3: Substitute the Known Value
From Step 1, we know that:
\[
\sin^2\theta - \cos^2\theta = 3
\]
Thus:
\[
\sin^4\theta - \cos^4\theta = 3
\]

### Final Answer
The value of \(\sin^4\theta - \cos^4\theta\) is \(\boxed{3}\).

Would you like to see any details or have any questions about this solution? 

Here are some additional related questions for further practice:
1. If \(\sin^2\theta - \cos^2\theta = 2\), what is the value of \(\sin^4\theta - \cos^4\theta\)?
2. Determine \(\sin^6\theta - \cos^6\theta\) if \(\sin^2\theta - \cos^2\theta = 4\).
3. Prove that \(\sin^4\theta + \cos^4\theta = 1 - 2\sin^2\theta\cos^2\theta\).
4. Solve for \(\theta\) given that \(\sin^2\theta - \cos^2\theta = 0\).
5. Find the maximum value of \(\sin^4\theta - \cos^4\theta\).

**Tip:** Whenever you encounter expressions involving trigonometric identities, look for ways to simplify them using known identities like the Pythagorean identity.

Learn how to solve the trigonometric expression sin^4(theta) - cos^4(theta) when given (sin(theta) + cos(theta))(sin(theta) - cos(theta)) = 3. This problem involves applying the Pythagorean identity and understanding trigonometric identities. Suitable for students in Grades 10-12.

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