https://file.aiquickdraw.com/mathbot/file/17259507573773xbpg9ak.jpg

The image shows the trigonometric equation:

\[
\cos^4\theta + \sin^4\theta = 1 - 2\cos^2\theta \sin^2\theta
\]

This is a well-known identity in trigonometry, and it can be derived using the identity \(\cos^2\theta + \sin^2\theta = 1\) and applying some algebraic manipulations.

### Derivation:
1. **Start with the given equation**:
   \[
   \cos^4\theta + \sin^4\theta
   \]

2. **Recognize that this can be written using the identity**:
   \[
   a^2 + b^2 = (a+b)^2 - 2ab
   \]
   Here, let \(a = \cos^2\theta\) and \(b = \sin^2\theta\), then:
   \[
   \cos^4\theta + \sin^4\theta = (\cos^2\theta + \sin^2\theta)^2 - 2\cos^2\theta\sin^2\theta
   \]

3. **Use the identity \(\cos^2\theta + \sin^2\theta = 1\)**:
   \[
   (\cos^2\theta + \sin^2\theta)^2 = 1^2 = 1
   \]
   So the expression becomes:
   \[
   \cos^4\theta + \sin^4\theta = 1 - 2\cos^2\theta\sin^2\theta
   \]

Thus, the equation is indeed correct.

Would you like more details on this identity or have any other questions?

### Related Questions:
1. What is the significance of the identity \(\cos^2\theta + \sin^2\theta = 1\) in trigonometry?
2. How can we use the identity \(\cos^2\theta + \sin^2\theta = 1\) to derive other trigonometric identities?
3. What are some applications of the identity \(\cos^4\theta + \sin^4\theta\) in solving trigonometric problems?
4. How can the double-angle identity \(\sin 2\theta = 2\sin\theta\cos\theta\) be used in related trigonometric problems?
5. Can the identity \(\cos^4\theta + \sin^4\theta\) be extended or generalized to other trigonometric identities?

### Tip:
When working with trigonometric identities, it’s often helpful to start by simplifying complex expressions using known identities, such as Pythagorean identities or angle sum and difference identities.

The user submitted a query regarding the trigonometric identity cos^4θ + sin^4θ = 1 - 2cos^2θsin^2θ.

Understand and derive the trigonometric identity cos^4θ + sin^4θ = 1 - 2cos^2θsin^2θ using the Pythagorean identity and algebraic manipulation. This problem is suitable for students in Grades 10-12.

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