Verify The Identity:$ \operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta $

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Introduction

In trigonometry, verifying identities is an essential skill that helps us simplify complex expressions and solve problems more efficiently. One of the most common identities in trigonometry is the Pythagorean identity, which states that $\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta = 1$. However, there are other identities that are equally important, and one of them is the identity we will be discussing in this article: $\operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$.

Understanding the Identity

Before we dive into verifying the identity, let's understand what it means. The left-hand side of the equation represents the sum of the fourth powers of sine and cosine, while the right-hand side represents a difference of squares. Our goal is to show that these two expressions are equivalent.

Step 1: Expand the Left-Hand Side

To verify the identity, we start by expanding the left-hand side of the equation. We can do this by using the binomial theorem or by simply multiplying out the terms.

$$\operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = (\operatorname{Sin}^2 \theta)^2 + (\operatorname{Cos}^2 \theta)^2$$

Using the binomial theorem, we can expand the squared terms as follows:

$$(\operatorname{Sin}^2 \theta)^2 = \operatorname{Sin}^4 \theta + 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta + \operatorname{Cos}^4 \theta$$

$$(\operatorname{Cos}^2 \theta)^2 = \operatorname{Cos}^4 \theta + 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta + \operatorname{Sin}^4 \theta$$

Step 2: Simplify the Left-Hand Side

Now that we have expanded the left-hand side, we can simplify it by combining like terms.

$$\operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = \operatorname{Sin}^4 \theta + 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta + \operatorname{Cos}^4 \theta + \operatorname{Cos}^4 \theta + 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta + \operatorname{Sin}^4 \theta$$

Combining like terms, we get:

$$\operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = 2 \operatorname{Sin}^4 \theta + 2 \operatorname{Cos}^4 \theta + 4 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$$

Step 3: Simplify the Right-Hand Side

Now that we have simplified the left-hand side, we can simplify the right-hand side of the equation.

$$1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$$

This expression is already simplified, so we can move on to the next step.

Step 4: Equate the Left-Hand Side and Right-Hand Side

Now that we have simplified both sides of the equation, we can equate them to verify the identity.

$$2 \operatorname{Sin}^4 \theta + 2 \operatorname{Cos}^4 \theta + 4 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$$

Step 5: Simplify the Equation

To simplify the equation, we can start by combining like terms.

$$2 \operatorname{Sin}^4 \theta + 2 \operatorname{Cos}^4 \theta + 4 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$$

$$2 \operatorname{Sin}^4 \theta + 2 \operatorname{Cos}^4 \theta + 6 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1$$

Step 6: Factor the Equation

To factor the equation, we can start by factoring out the common term.

$$2 \operatorname{Sin}^4 \theta + 2 \operatorname{Cos}^4 \theta + 6 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1$$

$$(\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta)^2 + 4 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1$$

Step 7: Simplify the Factored Equation

Now that we have factored the equation, we can simplify it by using the Pythagorean identity.

$$(\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta)^2 + 4 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1$$

$$(1)^2 + 4 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta = 1$$

Conclusion

In this article, we have verified the identity $\operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$. We started by expanding the left-hand side of the equation and then simplified it by combining like terms. We then simplified the right-hand side of the equation and equated the two expressions to verify the identity. Finally, we factored the equation and simplified it using the Pythagorean identity.

Final Answer

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Q: What is the purpose of verifying the identity $\operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$?

A: The purpose of verifying the identity is to show that the left-hand side of the equation is equivalent to the right-hand side. This is an important skill in trigonometry, as it allows us to simplify complex expressions and solve problems more efficiently.

Q: How do I verify the identity?

A: To verify the identity, you can follow these steps:

1. Expand the left-hand side of the equation using the binomial theorem or by simply multiplying out the terms.
2. Simplify the left-hand side by combining like terms.
3. Simplify the right-hand side of the equation.
4. Equate the left-hand side and right-hand side to verify the identity.
5. Factor the equation and simplify it using the Pythagorean identity.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is the equation $\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta = 1$. This identity is used to simplify expressions involving sine and cosine.

Q: How do I use the Pythagorean identity to simplify the equation?

A: To use the Pythagorean identity to simplify the equation, you can substitute the expression $(\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta)^2$ for $1$ in the equation.

Q: What is the final answer to the equation?

A: The final answer to the equation is $\boxed{1}$.

Q: Why is the final answer 1?

A: The final answer is 1 because the left-hand side of the equation is equivalent to the right-hand side. When we simplify the left-hand side, we get $2 \operatorname{Sin}^4 \theta + 2 \operatorname{Cos}^4 \theta + 4 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$. When we simplify the right-hand side, we get $1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$. Since the two expressions are equivalent, the final answer is 1.

Q: What are some common mistakes to avoid when verifying the identity?

A: Some common mistakes to avoid when verifying the identity include:

• Not expanding the left-hand side of the equation correctly
• Not simplifying the left-hand side of the equation correctly
• Not simplifying the right-hand side of the equation correctly
• Not equating the left-hand side and right-hand side correctly
• Not factoring the equation correctly

Q: How can I practice verifying the identity?

A: You can practice verifying the identity by working through example problems and exercises. You can also try verifying different identities to see how they work.

Q: What are some real-world applications of verifying the identity?

A: Verifying the identity has many real-world applications, including:

• Simplifying complex expressions in physics and engineering
• Solving problems in calculus and differential equations
• Working with trigonometric functions in computer science and programming
• Understanding the behavior of waves and oscillations in physics and engineering

Conclusion

In this article, we have answered some common questions about verifying the identity $\operatorname{Sin}^4 \theta + \operatorname{Cos}^4 \theta = 1 - 2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$. We have discussed the purpose of verifying the identity, how to verify the identity, and some common mistakes to avoid. We have also discussed some real-world applications of verifying the identity and how to practice verifying the identity.