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Let's analyze and solve both equations you provided:

### 1. \( \sin^4 \theta + \cos^4 \theta = 1 \)
This equation involves powers of sine and cosine. To simplify, notice that:

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

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we substitute this in:

\[
= 1^2 - 2 \sin^2 \theta \cos^2 \theta
= 1 - 2 \sin^2 \theta \cos^2 \theta
\]

So, the original equation becomes:

\[
1 - 2 \sin^2 \theta \cos^2 \theta = 1
\]

Now subtract 1 from both sides:

\[
-2 \sin^2 \theta \cos^2 \theta = 0
\]

Which implies:

\[
\sin^2 \theta \cos^2 \theta = 0
\]

This is true when either \( \sin \theta = 0 \) or \( \cos \theta = 0 \). Therefore, the solutions are:

\[
\theta = n\pi \quad \text{or} \quad \theta = \frac{\pi}{2} + n\pi, \quad \text{where } n \text{ is any integer}.
\]

### 2. \( \sin \theta (\cos \theta - 1) = \sin^2 \theta \)

Expand the left-hand side:

\[
\sin \theta \cos \theta - \sin \theta = \sin^2 \theta
\]

Now, rearrange the equation:

\[
\sin \theta \cos \theta - \sin^2 \theta = \sin \theta
\]

Factor out \( \sin \theta \) from the left-hand side:

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

If \( \sin \theta \neq 0 \), divide both sides by \( \sin \theta \):

\[
\cos \theta - \sin \theta = 1
\]

However, this equation doesn't have a standard simple solution. Therefore, the alternative case is when \( \sin \theta = 0 \). In this case, the solution is \( \theta = n\pi \), where \( n \) is any integer.

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Would you like more details on any of the steps, or do you have additional questions?

### Related questions:
1. Can you provide alternative methods for simplifying \( \sin^4 \theta + \cos^4 \theta \)?
2. How do we solve trigonometric equations using identities?
3. Can you explain how to derive the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \)?
4. What are the general solutions for \( \sin \theta = 0 \) and \( \cos \theta = 0 \)?
5. How does factoring help solve trigonometric equations?

### Tip:
When working with trigonometric equations, it's helpful to use known identities like the Pythagorean identity or product-to-sum identities to simplify expressions efficiently.

Solve the following trigonometric equations: 1. sin^4(θ) + cos^4(θ) = 1, 2. sin(θ)(cos(θ) - 1) = sin^2(θ)

Learn how to solve the trigonometric equations sin^4(θ) + cos^4(θ) = 1 and sin(θ)(cos(θ) - 1) = sin^2(θ) using Pythagorean identities and factoring techniques. Detailed step-by-step solutions provided. Suitable for high school students.

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