Simplify The Given Expression. Sin ⁡ ( 6 X ) Cos ⁡ ( 4 X ) − Sin ⁡ ( 4 X ) Cos ⁡ ( 6 X ) = \sin (6x) \cos (4x) - \sin (4x) \cos (6x) = Sin ( 6 X ) Cos ( 4 X ) − Sin ( 4 X ) Cos ( 6 X ) = □ \square □

Mar 12, 2025 by ADMIN 199 views

Simplify the Given Expression

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Introduction

In this article, we will simplify the given trigonometric expression involving sine and cosine functions. The expression is $\sin (6x) \cos (4x) - \sin (4x) \cos (6x)$ . We will use various trigonometric identities to simplify this expression and arrive at the final result.

Trigonometric Identities

Before we proceed with simplifying the given expression, let's recall some basic trigonometric identities that we will use:

Sine and Cosine Product-to-Sum Identity: $\sin A \cos B - \cos A \sin B = \sin (A - B)$
Cosine Double Angle Identity: $\cos 2A = 2\cos^2 A - 1$
Sine Double Angle Identity: $\sin 2A = 2\sin A \cos A$

Simplifying the Expression

Now, let's simplify the given expression using the trigonometric identities mentioned above.

$\sin (6x) \cos (4x) - \sin (4x) \cos (6x)$

Using the Sine and Cosine Product-to-Sum Identity, we can rewrite the expression as:

$\sin (6x - 4x) = \sin (2x)$

Therefore, the simplified expression is $\boxed{\sin (2x)}$ .

Conclusion

In this article, we simplified the given trigonometric expression involving sine and cosine functions. We used various trigonometric identities to arrive at the final result. The simplified expression is $\boxed{\sin (2x)}$ .

Example Use Cases

The simplified expression can be used in various mathematical and scientific applications, such as:

Trigonometry: The expression can be used to solve trigonometric equations and identities.
Calculus: The expression can be used to find derivatives and integrals of trigonometric functions.
Physics: The expression can be used to describe the motion of objects in terms of sine and cosine functions.

Final Answer

The final answer is $\boxed{\sin (2x)}$ .

Step-by-Step Solution

Here's a step-by-step solution to the problem:

Step 1: Recall the Sine and Cosine Product-to-Sum Identity: $\sin A \cos B - \cos A \sin B = \sin (A - B)$
Step 2: Apply the identity to the given expression: $\sin (6x) \cos (4x) - \sin (4x) \cos (6x) = \sin (6x - 4x)$
Step 3: Simplify the expression: $\sin (6x - 4x) = \sin (2x)$

Frequently Asked Questions

Here are some frequently asked questions related to the problem:

Q: What is the simplified expression? A: The simplified expression is $\boxed{\sin (2x)}$ .
Q: What trigonometric identities were used to simplify the expression? A: The Sine and Cosine Product-to-Sum Identity was used to simplify the expression.
Q: What are some example use cases of the simplified expression? A: The simplified expression can be used in various mathematical and scientific applications, such as trigonometry, calculus, and physics.

Introduction

In our previous article, we simplified the given trigonometric expression involving sine and cosine functions. The expression was $\sin (6x) \cos (4x) - \sin (4x) \cos (6x)$ . We used various trigonometric identities to simplify this expression and arrive at the final result. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the simplified expression?

A: The simplified expression is $\boxed{\sin (2x)}$ .

Q: What trigonometric identities were used to simplify the expression?

A: The Sine and Cosine Product-to-Sum Identity was used to simplify the expression.

Q: What are some example use cases of the simplified expression?

A: The simplified expression can be used in various mathematical and scientific applications, such as:

Trigonometry: The expression can be used to solve trigonometric equations and identities.
Calculus: The expression can be used to find derivatives and integrals of trigonometric functions.
Physics: The expression can be used to describe the motion of objects in terms of sine and cosine functions.

Q: How do I apply the Sine and Cosine Product-to-Sum Identity to simplify the expression?

A: To apply the identity, you need to identify the two sine and cosine functions in the expression and then use the identity to rewrite the expression.

Q: What are some common mistakes to avoid when simplifying trigonometric expressions?

A: Some common mistakes to avoid when simplifying trigonometric expressions include:

Not using the correct trigonometric identities: Make sure to use the correct trigonometric identities to simplify the expression.
Not simplifying the expression correctly: Make sure to simplify the expression correctly by following the steps outlined in the solution.
Not checking the final answer: Make sure to check the final answer to ensure that it is correct.

Q: How do I check the final answer?

A: To check the final answer, you can use various methods such as:

Substituting values: Substitute values into the expression to check if the final answer is correct.
Graphing: Graph the expression to check if the final answer is correct.
Using a calculator: Use a calculator to check if the final answer is correct.