Prove That $\frac{\cos \theta - \sin \theta \cdot \cos \theta}{\cos \theta + \sin \theta \cdot \cos \theta} = \frac{\operatorname{cosec} \theta - 1}{\operatorname{cosec} \theta + 1}$

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Introduction to Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable for which the functions are defined. These identities are used to simplify expressions, solve equations, and prove other identities. In this article, we will prove a trigonometric identity involving the cosine and cosecant functions.

The Given Identity

The given identity is cos⁡θ−sin⁡θ⋅cos⁡θcos⁡θ+sin⁡θ⋅cos⁡θ=cosec⁡θ−1cosec⁡θ+1\frac{\cos \theta - \sin \theta \cdot \cos \theta}{\cos \theta + \sin \theta \cdot \cos \theta} = \frac{\operatorname{cosec} \theta - 1}{\operatorname{cosec} \theta + 1}. To prove this identity, we will start by simplifying the left-hand side of the equation.

Simplifying the Left-Hand Side

We can simplify the left-hand side of the equation by factoring out the common term cos⁡θ\cos \theta from the numerator and denominator.

cos⁡θ−sin⁡θ⋅cos⁡θcos⁡θ+sin⁡θ⋅cos⁡θ=cos⁡θ(1−sin⁡θ)cos⁡θ(1+sin⁡θ)\frac{\cos \theta - \sin \theta \cdot \cos \theta}{\cos \theta + \sin \theta \cdot \cos \theta} = \frac{\cos \theta(1 - \sin \theta)}{\cos \theta(1 + \sin \theta)}

Canceling Common Terms

We can cancel out the common term cos⁡θ\cos \theta from the numerator and denominator.

cos⁡θ(1−sin⁡θ)cos⁡θ(1+sin⁡θ)=1−sin⁡θ1+sin⁡θ\frac{\cos \theta(1 - \sin \theta)}{\cos \theta(1 + \sin \theta)} = \frac{1 - \sin \theta}{1 + \sin \theta}

Simplifying the Right-Hand Side

We can simplify the right-hand side of the equation by using the definition of the cosecant function.

cosec⁡θ−1cosec⁡θ+1=1sin⁡θ−11sin⁡θ+1\frac{\operatorname{cosec} \theta - 1}{\operatorname{cosec} \theta + 1} = \frac{\frac{1}{\sin \theta} - 1}{\frac{1}{\sin \theta} + 1}

Simplifying the Right-Hand Side (Continued)

We can simplify the right-hand side of the equation by multiplying the numerator and denominator by sin⁡θ\sin \theta.

1sin⁡θ−11sin⁡θ+1=1−sin⁡θ1+sin⁡θ\frac{\frac{1}{\sin \theta} - 1}{\frac{1}{\sin \theta} + 1} = \frac{1 - \sin \theta}{1 + \sin \theta}

Conclusion

We have now simplified both sides of the given identity. We can see that the left-hand side and the right-hand side are equal, which proves the given identity.

Importance of Trigonometric Identities

Trigonometric identities are used in a wide range of applications, including physics, engineering, and mathematics. They are used to simplify expressions, solve equations, and prove other identities. In this article, we have proved a trigonometric identity involving the cosine and cosecant functions.

Real-World Applications of Trigonometric Identities

Trigonometric identities are used in a wide range of real-world applications, including:

  • Physics: Trigonometric identities are used to describe the motion of objects in terms of their position, velocity, and acceleration.
  • Engineering: Trigonometric identities are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
  • Mathematics: Trigonometric identities are used to prove other identities and to simplify expressions.

Conclusion

In this article, we have proved a trigonometric identity involving the cosine and cosecant functions. We have also discussed the importance of trigonometric identities and their real-world applications. Trigonometric identities are a fundamental part of mathematics and are used in a wide range of applications.

Final Thoughts

Trigonometric identities are a powerful tool for simplifying expressions, solving equations, and proving other identities. They are used in a wide range of applications, including physics, engineering, and mathematics. In this article, we have proved a trigonometric identity involving the cosine and cosecant functions, and we have discussed the importance of trigonometric identities and their real-world applications.

References

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak
  • "Mathematics for Engineers and Scientists" by Donald R. Hill

Further Reading

  • "Trigonometric Identities" by Paul Dawkins
  • "Trigonometry" by I. M. Gelfand
  • "Calculus" by James Stewart

Introduction

In our previous article, we proved a trigonometric identity involving the cosine and cosecant functions. In this article, we will answer some common questions related to proving trigonometric identities.

Q: What is a trigonometric identity?

A: A trigonometric identity is an equation that involves trigonometric functions and is true for all values of the variable for which the functions are defined.

Q: Why are trigonometric identities important?

A: Trigonometric identities are used to simplify expressions, solve equations, and prove other identities. They are a fundamental part of mathematics and are used in a wide range of applications, including physics, engineering, and mathematics.

Q: How do I prove a trigonometric identity?

A: To prove a trigonometric identity, you need to simplify both sides of the equation and show that they are equal. You can use a variety of techniques, including factoring, canceling common terms, and using trigonometric identities.

Q: What are some common techniques for proving trigonometric identities?

A: Some common techniques for proving trigonometric identities include:

  • Factoring: Factoring out common terms from the numerator and denominator.
  • Canceling common terms: Canceling out common terms from the numerator and denominator.
  • Using trigonometric identities: Using known trigonometric identities to simplify the expression.
  • Multiplying by a conjugate: Multiplying both sides of the equation by a conjugate to eliminate the radical.

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

A: To determine if a trigonometric identity is true, you need to simplify both sides of the equation and show that they are equal. You can use a variety of techniques, including factoring, canceling common terms, and using trigonometric identities.

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

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

  • Not simplifying both sides of the equation: Make sure to simplify both sides of the equation to show that they are equal.
  • Not using the correct trigonometric identities: Make sure to use the correct trigonometric identities to simplify the expression.
  • Not canceling common terms: Make sure to cancel out common terms from the numerator and denominator.

Q: How do I apply trigonometric identities in real-world problems?

A: Trigonometric identities are used in a wide range of real-world applications, including physics, engineering, and mathematics. To apply trigonometric identities in real-world problems, you need to:

  • Understand the problem: Understand the problem and the trigonometric functions involved.
  • Choose the correct trigonometric identity: Choose the correct trigonometric identity to simplify the expression.
  • Simplify the expression: Simplify the expression using the chosen trigonometric identity.

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

A: Some common real-world applications of trigonometric identities include:

  • Physics: Trigonometric identities are used to describe the motion of objects in terms of their position, velocity, and acceleration.
  • Engineering: Trigonometric identities are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
  • Mathematics: Trigonometric identities are used to prove other identities and to simplify expressions.

Conclusion

In this article, we have answered some common questions related to proving trigonometric identities. We have discussed the importance of trigonometric identities, common techniques for proving them, and common mistakes to avoid. We have also discussed how to apply trigonometric identities in real-world problems and some common real-world applications of trigonometric identities.

Final Thoughts

Trigonometric identities are a powerful tool for simplifying expressions, solving equations, and proving other identities. They are used in a wide range of applications, including physics, engineering, and mathematics. In this article, we have provided a comprehensive guide to proving trigonometric identities and applying them in real-world problems.

References

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak
  • "Mathematics for Engineers and Scientists" by Donald R. Hill

Further Reading

  • "Trigonometric Identities" by Paul Dawkins
  • "Trigonometry" by I. M. Gelfand
  • "Calculus" by James Stewart