Verify The Trigonometric Identity: Cos ⁡ 5 X − Cos ⁡ 15 X = 2 Sin ⁡ 10 X Sin ⁡ 5 X \cos 5x - \cos 15x = 2 \sin 10x \sin 5x Cos 5 X − Cos 15 X = 2 Sin 10 X Sin 5 X

$$

Introduction

Trigonometric identities are fundamental concepts in mathematics, particularly in trigonometry. These identities help us simplify complex trigonometric expressions and solve various mathematical problems. In this article, we will focus on verifying the trigonometric identity: $\cos 5x - \cos 15x = 2 \sin 10x \sin 5x$. This identity involves the cosine and sine functions, and it is essential to understand the properties of these functions to verify the given identity.

Understanding the Trigonometric Functions

Before we dive into verifying the identity, let's briefly review the trigonometric functions involved. The cosine function, denoted by $\cos x$, is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The sine function, denoted by $\sin x$, is also a periodic function that oscillates between -1 and 1. It is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Using the Sum-to-Product Identities

To verify the given identity, we can use the sum-to-product identities for cosine and sine functions. The sum-to-product identity for cosine is given by:

$$\cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$$

Similarly, the sum-to-product identity for sine is given by:

$$\sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$$

Verifying the Identity

Now, let's apply the sum-to-product identity for cosine to the given expression:

$$\cos 5x - \cos 15x = -2 \sin \left(\frac{5x+15x}{2}\right) \sin \left(\frac{5x-15x}{2}\right)$$

Simplifying the expression, we get:

$$\cos 5x - \cos 15x = -2 \sin 10x \sin (-5x)$$

Since $\sin (-x) = -\sin x$, we can rewrite the expression as:

$$\cos 5x - \cos 15x = 2 \sin 10x \sin 5x$$

Conclusion

In this article, we verified the trigonometric identity: $\cos 5x - \cos 15x = 2 \sin 10x \sin 5x$. We used the sum-to-product identities for cosine and sine functions to simplify the given expression. This identity is essential in trigonometry, and understanding its properties can help us solve various mathematical problems.

Applications of the Identity

The verified identity has several applications in mathematics and physics. For example, it can be used to simplify complex trigonometric expressions in calculus and differential equations. In physics, it can be used to describe the motion of objects in terms of their angular displacement and velocity.

Final Thoughts

Verifying trigonometric identities is an essential skill in mathematics, particularly in trigonometry. By understanding the properties of trigonometric functions and using sum-to-product identities, we can simplify complex expressions and solve various mathematical problems. The verified identity in this article is a fundamental concept in trigonometry, and it has several applications in mathematics and physics.

Additional Resources

For further reading on trigonometric identities and their applications, we recommend the following resources:

• "Trigonometry" by Michael Corral
• "Calculus" by Michael Spivak
• "Differential Equations" by Lawrence Perko

Frequently Asked Questions

Q: What is the sum-to-product identity for cosine? A: The sum-to-product identity for cosine is given by: $\cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$

Q: How can I use the sum-to-product identity to verify the given identity? A: You can use the sum-to-product identity for cosine to simplify the given expression: $\cos 5x - \cos 15x = -2 \sin 10x \sin (-5x)$

Q: What are the applications of the verified identity? A: The verified identity has several applications in mathematics and physics, including simplifying complex trigonometric expressions in calculus and differential equations, and describing the motion of objects in terms of their angular displacement and velocity.

$$

Introduction

In our previous article, we verified the trigonometric identity: $\cos 5x - \cos 15x = 2 \sin 10x \sin 5x$. This identity is a fundamental concept in trigonometry, and understanding its properties can help us solve various mathematical problems. In this article, we will provide a Q&A section to help you better understand the identity and its applications.

Q&A Section

Q: What is the sum-to-product identity for cosine?

A: The sum-to-product identity for cosine is given by: $\cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$

Q: How can I use the sum-to-product identity to verify the given identity?

A: You can use the sum-to-product identity for cosine to simplify the given expression: $\cos 5x - \cos 15x = -2 \sin 10x \sin (-5x)$. Since $\sin (-x) = -\sin x$, we can rewrite the expression as: $\cos 5x - \cos 15x = 2 \sin 10x \sin 5x$

Q: What are the applications of the verified identity?

A: The verified identity has several applications in mathematics and physics, including simplifying complex trigonometric expressions in calculus and differential equations, and describing the motion of objects in terms of their angular displacement and velocity.

Q: Can I use the verified identity to solve problems in physics?

A: Yes, the verified identity can be used to solve problems in physics, particularly in the context of rotational motion and angular displacement.

Q: How can I apply the verified identity to simplify complex trigonometric expressions?

A: You can use the verified identity to simplify complex trigonometric expressions by substituting the given expression with the simplified form: $\cos 5x - \cos 15x = 2 \sin 10x \sin 5x$

Q: What are some common mistakes to avoid when verifying trigonometric identities?

A: Some common mistakes to avoid when verifying trigonometric identities include:

• Not using the correct sum-to-product identity
• Not simplifying the expression correctly
• Not checking the validity of the identity

Q: How can I check the validity of a trigonometric identity?

A: You can check the validity of a trigonometric identity by:

• Using the sum-to-product identity to simplify the expression
• Checking the expression against the original identity
• Verifying that the simplified expression is equivalent to the original expression

Q: What are some real-world applications of trigonometric identities?

A: Trigonometric identities have numerous real-world applications, including:

• Navigation and surveying
• Physics and engineering
• Computer graphics and animation
• Medical imaging and diagnostics

Conclusion

In this Q&A article, we provided answers to common questions about the trigonometric identity: $\cos 5x - \cos 15x = 2 \sin 10x \sin 5x$. We hope that this article has helped you better understand the identity and its applications. If you have any further questions or need additional clarification, please don't hesitate to ask.

Additional Resources

For further reading on trigonometric identities and their applications, we recommend the following resources:

• "Trigonometry" by Michael Corral
• "Calculus" by Michael Spivak
• "Differential Equations" by Lawrence Perko

Final Thoughts

Verifying trigonometric identities is an essential skill in mathematics, particularly in trigonometry. By understanding the properties of trigonometric functions and using sum-to-product identities, we can simplify complex expressions and solve various mathematical problems. The verified identity in this article is a fundamental concept in trigonometry, and it has several applications in mathematics and physics.