Solve Trignometric Identities. 1/1+costheta + 1/1-costheta = 2+2cot²theta

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Introduction

Trigonometric identities are equations that are true for all values of the variables involved. They are used to simplify expressions and solve equations in trigonometry. In this article, we will solve the trigonometric identity: 1/1+costheta + 1/1-costheta = 2+2cot²theta.

Understanding the Identity

The given identity is: 1/1+costheta + 1/1-costheta = 2+2cot²theta. To solve this identity, we need to simplify the left-hand side and show that it is equal to the right-hand side.

Step 1: Simplify the Left-Hand Side

We can start by simplifying the left-hand side of the identity. We can combine the two fractions on the left-hand side by finding a common denominator.

1/(1+costheta) + 1/(1-costheta) = (1+costheta + 1-costheta) / ((1+costheta)(1-costheta))

Step 2: Simplify the Denominator

We can simplify the denominator by using the difference of squares formula: (a+b)(a-b) = a² - b².

(1+costheta)(1-costheta) = 1² - (costheta)²

Step 3: Simplify the Numerator

We can simplify the numerator by combining like terms.

1+costheta + 1-costheta = 2

Step 4: Simplify the Left-Hand Side

Now we can simplify the left-hand side by substituting the simplified numerator and denominator.

(1+costheta + 1-costheta) / ((1+costheta)(1-costheta)) = 2 / (1 - (costheta)²)

Step 5: Simplify the Right-Hand Side

We can simplify the right-hand side by using the definition of cotangent: cotθ = cosθ / sinθ.

2+2cot²theta = 2 + 2(cosθ / sinθ)²

Step 6: Simplify the Right-Hand Side

We can simplify the right-hand side by using the identity: (a/b)² = a² / b².

2 + 2(cosθ / sinθ)² = 2 + 2(cos²θ / sin²θ)

Step 7: Simplify the Right-Hand Side

We can simplify the right-hand side by using the identity: cos²θ + sin²θ = 1.

2 + 2(cos²θ / sin²θ) = 2 + 2(1 - sin²θ) / sin²θ

Step 8: Simplify the Right-Hand Side

We can simplify the right-hand side by combining like terms.

2 + 2(1 - sin²θ) / sin²θ = 2 + 2(1 / sin²θ) - 2

Step 9: Simplify the Right-Hand Side

We can simplify the right-hand side by combining like terms.

2 + 2(1 / sin²θ) - 2 = 2(1 / sin²θ)

Step 10: Simplify the Right-Hand Side

We can simplify the right-hand side by using the identity: 1 / sin²θ = cot²θ.

2(1 / sin²θ) = 2cot²θ

Conclusion

We have shown that the given trigonometric identity: 1/1+costheta + 1/1-costheta = 2+2cot²theta is true. We simplified the left-hand side by combining the two fractions and simplifying the denominator. We then simplified the right-hand side by using the definition of cotangent and the identity: cos²θ + sin²θ = 1.

Final Answer

The final answer is: 2+2cot²theta.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Trigonometric Identities" by Paul Dawkins

Note

Introduction

In our previous article, we solved the trigonometric identity: 1/1+costheta + 1/1-costheta = 2+2cot²theta. In this article, we will answer some common questions related to trigonometric identities and provide additional examples.

Q: What is a trigonometric identity?

A: A trigonometric identity is an equation that is true for all values of the variables involved. They are used to simplify expressions and solve equations in trigonometry.

Q: How do I know if a trigonometric identity is true?

A: To determine if a trigonometric identity is true, you can use various methods such as:

  • Simplifying both sides of the equation
  • Using trigonometric identities and formulas
  • Graphing the functions and checking if they are equal
  • Using algebraic manipulations

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin²θ + cos²θ = 1
  • tan²θ + 1 = sec²θ
  • cot²θ + 1 = cosec²θ
  • sin(2θ) = 2sinθcosθ
  • cos(2θ) = cos²θ - sin²θ

Q: How do I simplify a trigonometric expression?

A: To simplify a trigonometric expression, you can use various methods such as:

  • Combining like terms
  • Using trigonometric identities and formulas
  • Factoring out common factors
  • Using algebraic manipulations

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

A: A trigonometric identity is an equation that is true for all values of the variables involved, while a trigonometric equation is an equation that is true for specific values of the variables involved.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can use various methods such as:

  • Using trigonometric identities and formulas
  • Graphing the functions and finding the intersection points
  • Using algebraic manipulations
  • Using numerical methods

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

A: Some common mistakes to avoid when working with trigonometric identities include:

  • Not simplifying both sides of the equation
  • Not using trigonometric identities and formulas correctly
  • Not checking if the equation is true for all values of the variables involved
  • Not using algebraic manipulations correctly

Q: How do I know if a trigonometric identity is true for all values of the variables involved?

A: To determine if a trigonometric identity is true for all values of the variables involved, you can use various methods such as:

  • Simplifying both sides of the equation
  • Using trigonometric identities and formulas
  • Graphing the functions and checking if they are equal
  • Using algebraic manipulations

Conclusion

In this article, we answered some common questions related to trigonometric identities and provided additional examples. We also discussed some common mistakes to avoid when working with trigonometric identities.

Final Answer

The final answer is: Trigonometric identities are an essential part of trigonometry and are used to simplify expressions and solve equations.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Trigonometric Identities" by Paul Dawkins

Note

This article is for educational purposes only. The author is not responsible for any errors or inaccuracies in the article.