What Is The Simplest Form Of The Expression Below?$\[ \frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) - \frac{\sin (\theta)}{\cos (\theta) \tan (\theta)} \\]A. \[$\sec \theta\$\]B. \[$\cot \theta\$\]C.

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Introduction

Trigonometric expressions can be complex and daunting, but with the right approach, they can be simplified to reveal their underlying beauty. In this article, we will explore the simplest form of a given expression, using a combination of trigonometric identities and algebraic manipulation.

The Given Expression

The expression we will be simplifying is:

cot⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)−sin⁡(θ)cos⁡(θ)tan⁡(θ)\frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) - \frac{\sin (\theta)}{\cos (\theta) \tan (\theta)}

Step 1: Simplify the First Term

To simplify the first term, we can start by using the identity cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}. We can rewrite the first term as:

cot⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)=cos⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)\frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) = \frac{\cos (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta)

Using the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}, we can rewrite the first term as:

cos⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)=cos⁡(θ)cos⁡(θ)sin⁡(θ)⋅sin⁡(θ)cos⁡(θ)\frac{\cos (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) = \frac{\cos (\theta) \cos (\theta)}{\sin (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)}

Simplifying further, we get:

cos⁡(θ)cos⁡(θ)sin⁡(θ)⋅sin⁡(θ)cos⁡(θ)=cos⁡(θ)\frac{\cos (\theta) \cos (\theta)}{\sin (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)} = \cos (\theta)

Step 2: Simplify the Second Term

To simplify the second term, we can start by using the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}. We can rewrite the second term as:

sin⁡(θ)cos⁡(θ)tan⁡(θ)=sin⁡(θ)cos⁡(θ)sin⁡(θ)cos⁡(θ)\frac{\sin (\theta)}{\cos (\theta) \tan (\theta)} = \frac{\sin (\theta)}{\cos (\theta) \frac{\sin (\theta)}{\cos (\theta)}}

Simplifying further, we get:

sin⁡(θ)cos⁡(θ)sin⁡(θ)cos⁡(θ)=sin⁡(θ)sin⁡(θ)=1\frac{\sin (\theta)}{\cos (\theta) \frac{\sin (\theta)}{\cos (\theta)}} = \frac{\sin (\theta)}{\sin (\theta)} = 1

Step 3: Combine the Terms

Now that we have simplified both terms, we can combine them to get the final expression:

cos⁡(θ)−1\cos (\theta) - 1

Conclusion

In this article, we simplified a complex trigonometric expression using a combination of trigonometric identities and algebraic manipulation. We started by simplifying the first term using the identity cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}, and then simplified the second term using the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}. Finally, we combined the two terms to get the final expression: cos⁡(θ)−1\cos (\theta) - 1.

Answer

The simplest form of the given expression is:

cos⁡(θ)−1\cos (\theta) - 1

This is option C.

Discussion

This problem requires a good understanding of trigonometric identities and algebraic manipulation. The student should be able to recognize the identities and apply them correctly to simplify the expression. The student should also be able to combine the terms to get the final expression.

Tips and Tricks

  • Make sure to recognize the trigonometric identities and apply them correctly.
  • Use algebraic manipulation to simplify the expression.
  • Combine the terms to get the final expression.

Practice Problems

  1. Simplify the expression: sin⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)−cos⁡(θ)sin⁡(θ)tan⁡(θ)\frac{\sin (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) - \frac{\cos (\theta)}{\sin (\theta) \tan (\theta)}
  2. Simplify the expression: cot⁡(θ)sin⁡(θ)cos⁡(θ)⋅tan⁡(θ)−cos⁡(θ)sin⁡(θ)tan⁡(θ)\frac{\cot (\theta) \sin (\theta)}{\cos (\theta)} \cdot \tan (\theta) - \frac{\cos (\theta)}{\sin (\theta) \tan (\theta)}

Answer Key

  1. cos⁡(θ)−1\cos (\theta) - 1
  2. cot⁡(θ)−1\cot (\theta) - 1

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Algebra and Trigonometry" by James Stewart

Q: What is the simplest form of the expression cot⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)−sin⁡(θ)cos⁡(θ)tan⁡(θ)\frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) - \frac{\sin (\theta)}{\cos (\theta) \tan (\theta)}?

A: The simplest form of the expression is cos⁡(θ)−1\cos (\theta) - 1.

Q: How do I simplify the first term of the expression?

A: To simplify the first term, you can start by using the identity cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}. You can rewrite the first term as cos⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)\frac{\cos (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta). Using the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}, you can rewrite the first term as cos⁡(θ)cos⁡(θ)sin⁡(θ)⋅sin⁡(θ)cos⁡(θ)\frac{\cos (\theta) \cos (\theta)}{\sin (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)}. Simplifying further, you get cos⁡(θ)\cos (\theta).

Q: How do I simplify the second term of the expression?

A: To simplify the second term, you can start by using the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}. You can rewrite the second term as sin⁡(θ)cos⁡(θ)tan⁡(θ)=sin⁡(θ)cos⁡(θ)sin⁡(θ)cos⁡(θ)\frac{\sin (\theta)}{\cos (\theta) \tan (\theta)} = \frac{\sin (\theta)}{\cos (\theta) \frac{\sin (\theta)}{\cos (\theta)}}. Simplifying further, you get sin⁡(θ)sin⁡(θ)=1\frac{\sin (\theta)}{\sin (\theta)} = 1.

Q: What is the final expression after simplifying both terms?

A: The final expression after simplifying both terms is cos⁡(θ)−1\cos (\theta) - 1.

Q: What trigonometric identities are used in this problem?

A: The trigonometric identities used in this problem are:

  • cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}
  • tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}

Q: How do I apply these identities to simplify the expression?

A: To apply these identities, you can start by rewriting the expression using the identities. For example, you can rewrite the first term as cos⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)\frac{\cos (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta). Then, you can simplify the expression using algebraic manipulation.

Q: What are some tips and tricks for simplifying trigonometric expressions?

A: Some tips and tricks for simplifying trigonometric expressions include:

  • Recognizing trigonometric identities and applying them correctly
  • Using algebraic manipulation to simplify the expression
  • Combining terms to get the final expression

Q: What are some practice problems for simplifying trigonometric expressions?

A: Some practice problems for simplifying trigonometric expressions include:

  • Simplify the expression: sin⁡(θ)cos⁡(θ)sin⁡(θ)⋅tan⁡(θ)−cos⁡(θ)sin⁡(θ)tan⁡(θ)\frac{\sin (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) - \frac{\cos (\theta)}{\sin (\theta) \tan (\theta)}
  • Simplify the expression: cot⁡(θ)sin⁡(θ)cos⁡(θ)⋅tan⁡(θ)−cos⁡(θ)sin⁡(θ)tan⁡(θ)\frac{\cot (\theta) \sin (\theta)}{\cos (\theta)} \cdot \tan (\theta) - \frac{\cos (\theta)}{\sin (\theta) \tan (\theta)}

Q: What are some resources for learning more about trigonometric expressions?

A: Some resources for learning more about trigonometric expressions include:

  • "Trigonometry" by Michael Corral
  • "Algebra and Trigonometry" by James Stewart

Note: The resources provided are for general information and are not specific to this problem.