Which Of The Following Is A Simpler Form Of The Expression Sin ⁡ ( Θ ) Sec ⁡ ( Θ ) Cos ⁡ ( Θ ) Tan ⁡ ( Θ ) \frac{\sin (\theta) \sec (\theta)}{\cos (\theta) \tan (\theta)} C O S ( Θ ) T A N ( Θ ) S I N ( Θ ) S E C ( Θ ) ​ ?A. Sec ⁡ Θ \sec \theta Sec Θ B. Cos ⁡ Θ \cos \theta Cos Θ C. Csc ⁡ Θ \csc \theta Csc Θ D. Tan ⁡ Θ \tan \theta Tan Θ

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Introduction

Trigonometric expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying beauty. In this article, we will explore the process of simplifying a specific trigonometric expression, sin⁡(θ)sec⁡(θ)cos⁡(θ)tan⁡(θ)\frac{\sin (\theta) \sec (\theta)}{\cos (\theta) \tan (\theta)}, and determine which of the given options is a simpler form of this expression.

Understanding the Expression

Before we dive into simplifying the expression, let's break it down and understand its components. The expression is a fraction, with the numerator being the product of sin⁡(θ)\sin (\theta) and sec⁡(θ)\sec (\theta), and the denominator being the product of cos⁡(θ)\cos (\theta) and tan⁡(θ)\tan (\theta).

Reciprocal Identities

To simplify this expression, we need to use reciprocal identities. The reciprocal identity for sine is csc⁡(θ)=1sin⁡(θ)\csc (\theta) = \frac{1}{\sin (\theta)}, and the reciprocal identity for cosine is sec⁡(θ)=1cos⁡(θ)\sec (\theta) = \frac{1}{\cos (\theta)}. We can also use the reciprocal identity for tangent, which is cot⁡(θ)=1tan⁡(θ)\cot (\theta) = \frac{1}{\tan (\theta)}.

Simplifying the Expression

Now that we have a good understanding of the expression and the reciprocal identities, let's simplify it step by step.

Step 1: Simplify the Numerator

The numerator of the expression is sin⁡(θ)sec⁡(θ)\sin (\theta) \sec (\theta). We can simplify this by using the reciprocal identity for cosine, which is sec⁡(θ)=1cos⁡(θ)\sec (\theta) = \frac{1}{\cos (\theta)}. Therefore, the numerator becomes sin⁡(θ)1cos⁡(θ)\sin (\theta) \frac{1}{\cos (\theta)}.

Step 2: Simplify the Denominator

The denominator of the expression is cos⁡(θ)tan⁡(θ)\cos (\theta) \tan (\theta). We can simplify this by using the reciprocal identity for tangent, which is tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}. Therefore, the denominator becomes cos⁡(θ)sin⁡(θ)cos⁡(θ)\cos (\theta) \frac{\sin (\theta)}{\cos (\theta)}.

Step 3: Cancel Out Common Factors

Now that we have simplified the numerator and denominator, we can cancel out common factors. The numerator is sin⁡(θ)1cos⁡(θ)\sin (\theta) \frac{1}{\cos (\theta)}, and the denominator is cos⁡(θ)sin⁡(θ)cos⁡(θ)\cos (\theta) \frac{\sin (\theta)}{\cos (\theta)}. We can cancel out the cos⁡(θ)\cos (\theta) term in the numerator and denominator, leaving us with sin⁡(θ)sin⁡(θ)\frac{\sin (\theta)}{\sin (\theta)}.

Step 4: Simplify the Expression

The expression sin⁡(θ)sin⁡(θ)\frac{\sin (\theta)}{\sin (\theta)} is equal to 1, since any non-zero number divided by itself is equal to 1.

Conclusion

In conclusion, the simplified form of the expression sin⁡(θ)sec⁡(θ)cos⁡(θ)tan⁡(θ)\frac{\sin (\theta) \sec (\theta)}{\cos (\theta) \tan (\theta)} is 1. This is a much simpler form than the original expression, and it reveals the underlying beauty of the trigonometric functions.

Answer

The correct answer is A. sec⁡θ\sec \theta. This is because the simplified form of the expression is 1, which is equal to sec⁡θ\sec \theta.

Final Thoughts

Q: What is the main goal of simplifying trigonometric expressions?

A: The main goal of simplifying trigonometric expressions is to reduce complex expressions to their simplest form, making it easier to work with and understand the underlying relationships between the trigonometric functions.

Q: What are some common techniques used to simplify trigonometric expressions?

A: Some common techniques used to simplify trigonometric expressions include:

  • Using reciprocal identities
  • Canceling out common factors
  • Applying trigonometric identities
  • Using algebraic manipulations

Q: How do I know when to use reciprocal identities?

A: You should use reciprocal identities when you see a trigonometric function in the denominator of an expression. For example, if you see 1sin⁡(θ)\frac{1}{\sin (\theta)}, you can use the reciprocal identity for sine, which is csc⁡(θ)=1sin⁡(θ)\csc (\theta) = \frac{1}{\sin (\theta)}.

Q: What is the difference between a trigonometric identity and a reciprocal identity?

A: A trigonometric identity is a statement that two or more trigonometric expressions are equal. For example, the Pythagorean identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2 (\theta) + \cos^2 (\theta) = 1 is a trigonometric identity. A reciprocal identity, on the other hand, is a specific type of trigonometric identity that relates a trigonometric function to its reciprocal. For example, the reciprocal identity for sine is csc⁡(θ)=1sin⁡(θ)\csc (\theta) = \frac{1}{\sin (\theta)}.

Q: How do I know when to cancel out common factors?

A: You should cancel out common factors when you see two or more expressions that have a common factor. For example, if you see sin⁡(θ)cos⁡(θ)cos⁡(θ)sin⁡(θ)\frac{\sin (\theta) \cos (\theta)}{\cos (\theta) \sin (\theta)}, you can cancel out the cos⁡(θ)\cos (\theta) term in the numerator and denominator, leaving you with sin⁡(θ)sin⁡(θ)\frac{\sin (\theta)}{\sin (\theta)}.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

  • The Pythagorean identity: sin⁡2(θ)+cos⁡2(θ)=1\sin^2 (\theta) + \cos^2 (\theta) = 1
  • The reciprocal identities: csc⁡(θ)=1sin⁡(θ)\csc (\theta) = \frac{1}{\sin (\theta)}, sec⁡(θ)=1cos⁡(θ)\sec (\theta) = \frac{1}{\cos (\theta)}, and cot⁡(θ)=1tan⁡(θ)\cot (\theta) = \frac{1}{\tan (\theta)}
  • The sum and difference identities: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A + B) = \sin A \cos B + \cos A \sin B and cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A \cos B - \sin A \sin B

Q: How do I apply trigonometric identities to simplify expressions?

A: To apply trigonometric identities to simplify expressions, you should:

  1. Identify the trigonometric functions in the expression
  2. Determine which trigonometric identity to use
  3. Apply the identity to simplify the expression
  4. Check your work to make sure the expression is simplified correctly

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 reciprocal identities when necessary
  • Not canceling out common factors when possible
  • Not applying trigonometric identities correctly
  • Not checking your work to make sure the expression is simplified correctly

Conclusion

Simplifying trigonometric expressions can be a challenging task, but with the right techniques and strategies, it can be done. By using reciprocal identities, canceling out common factors, and applying trigonometric identities, you can simplify complex expressions and reveal the underlying beauty of the trigonometric functions. Remember to always check your work to make sure the expression is simplified correctly.