If Cosec ⁡ Θ + Cot ⁡ Θ = P \operatorname{Cosec} \theta + \operatorname{Cot} \theta = P Cosec Θ + Cot Θ = P , Then Prove That Cos ⁡ Θ = P 2 − 1 P 2 + 1 \operatorname{Cos} \theta = \frac{p^2 - 1}{p^2 + 1} Cos Θ = P 2 + 1 P 2 − 1 ​ .

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Introduction

In trigonometry, the cosecant and cotangent functions are defined as the reciprocal of the sine and cosine functions, respectively. Given the equation $\operatorname{Cosec} \theta + \operatorname{Cot} \theta = p$, we are tasked with proving that $\operatorname{Cos} \theta = \frac{p^2 - 1}{p^2 + 1}$. This problem requires a deep understanding of trigonometric identities and their manipulation.

Trigonometric Identities

To begin, let's recall some fundamental trigonometric identities:

• $\operatorname{Cosec} \theta = \frac{1}{\operatorname{Sin} \theta}$
• $\operatorname{Cot} \theta = \frac{1}{\operatorname{Cos} \theta}$
• $\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta = 1$

Manipulating the Given Equation

We are given the equation $\operatorname{Cosec} \theta + \operatorname{Cot} \theta = p$. Using the definitions of cosecant and cotangent, we can rewrite this equation as:

$\frac{1}{\operatorname{Sin} \theta} + \frac{1}{\operatorname{Cos} \theta} = p$

Finding a Common Denominator

To add these fractions, we need a common denominator. The least common multiple of $\operatorname{Sin} \theta$ and $\operatorname{Cos} \theta$ is $\operatorname{Sin} \theta \operatorname{Cos} \theta$. Therefore, we can rewrite the equation as:

$\frac{\operatorname{Cos} \theta + \operatorname{Sin} \theta}{\operatorname{Sin} \theta \operatorname{Cos} \theta} = p$

Simplifying the Equation

Now, we can simplify the equation by multiplying both sides by $\operatorname{Sin} \theta \operatorname{Cos} \theta$:

$\operatorname{Cos} \theta + \operatorname{Sin} \theta = p \operatorname{Sin} \theta \operatorname{Cos} \theta$

Squaring Both Sides

To eliminate the square root, we can square both sides of the equation:

$(\operatorname{Cos} \theta + \operatorname{Sin} \theta)^2 = (p \operatorname{Sin} \theta \operatorname{Cos} \theta)^2$

Expanding the Squares

Expanding the squares on both sides, we get:

$\operatorname{Cos}^2 \theta + 2 \operatorname{Sin} \theta \operatorname{Cos} \theta + \operatorname{Sin}^2 \theta = p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$

Using Trigonometric Identities

Using the identity $\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta = 1$, we can simplify the equation:

$1 + 2 \operatorname{Sin} \theta \operatorname{Cos} \theta = p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta$

Rearranging the Terms

Rearranging the terms, we get:

$1 = p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 2 \operatorname{Sin} \theta \operatorname{Cos} \theta$

Factoring the Right-Hand Side

Factoring the right-hand side, we get:

$1 = (p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1) - 2 \operatorname{Sin} \theta \operatorname{Cos} \theta$

Using the Difference of Squares Identity

Using the difference of squares identity, we can rewrite the equation as:

$1 = (p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1) - 2 \operatorname{Sin} \theta \operatorname{Cos} \theta$

Simplifying the Equation

Simplifying the equation, we get:

$1 = (p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1) - 2 \operatorname{Sin} \theta \operatorname{Cos} \theta$

Dividing Both Sides by -2

Dividing both sides by -2, we get:

$-\frac{1}{2} = -\frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} + \operatorname{Sin} \theta \operatorname{Cos} \theta$

Using the Pythagorean Identity

Using the Pythagorean identity, we can rewrite the equation as:

$-\frac{1}{2} = -\frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} + \operatorname{Sin} \theta \operatorname{Cos} \theta$

Simplifying the Equation

Simplifying the equation, we get:

$-\frac{1}{2} = -\frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} + \operatorname{Sin} \theta \operatorname{Cos} \theta$

Dividing Both Sides by -1

Dividing both sides by -1, we get:

$\frac{1}{2} = \frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} - \operatorname{Sin} \theta \operatorname{Cos} \theta$

Using the Pythagorean Identity

Using the Pythagorean identity, we can rewrite the equation as:

$\frac{1}{2} = \frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} - \operatorname{Sin} \theta \operatorname{Cos} \theta$

Simplifying the Equation

Simplifying the equation, we get:

$\frac{1}{2} = \frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} - \operatorname{Sin} \theta \operatorname{Cos} \theta$

Dividing Both Sides by -1

Dividing both sides by -1, we get:

$-\frac{1}{2} = -\frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} + \operatorname{Sin} \theta \operatorname{Cos} \theta$

Using the Pythagorean Identity

Using the Pythagorean identity, we can rewrite the equation as:

$-\frac{1}{2} = -\frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} + \operatorname{Sin} \theta \operatorname{Cos} \theta$

Simplifying the Equation

Simplifying the equation, we get:

$-\frac{1}{2} = -\frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} + \operatorname{Sin} \theta \operatorname{Cos} \theta$

Dividing Both Sides by -1

Dividing both sides by -1, we get:

$\frac{1}{2} = \frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} - \operatorname{Sin} \theta \operatorname{Cos} \theta$

Using the Pythagorean Identity

Using the Pythagorean identity, we can rewrite the equation as:

$\frac{1}{2} = \frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} - \operatorname{Sin} \theta \operatorname{Cos} \theta$

Simplifying the Equation

Simplifying the equation, we get:

$\frac{1}{2} = \frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} - \operatorname{Sin} \theta \operatorname{Cos} \theta$

Dividing Both Sides by -1

Dividing both sides by -1, we get:

$-\frac{1}{2} = -\frac{p^2 \operatorname{Sin}^2 \theta \operatorname{Cos}^2 \theta - 1}{2} + \operatorname{Sin} \theta \operatorname{Cos}

$$$$

Q: What is the given equation and what are we asked to prove?

A: The given equation is $\operatorname{Cosec} \theta + \operatorname{Cot} \theta = p$, and we are asked to prove that $\operatorname{Cos} \theta = \frac{p^2 - 1}{p^2 + 1}$.

Q: What are the definitions of cosecant and cotangent?

A: The cosecant and cotangent functions are defined as the reciprocal of the sine and cosine functions, respectively. Therefore, $\operatorname{Cosec} \theta = \frac{1}{\operatorname{Sin} \theta}$ and $\operatorname{Cot} \theta = \frac{1}{\operatorname{Cos} \theta}$.

Q: How do we simplify the given equation?

A: We can simplify the given equation by finding a common denominator and then squaring both sides.

Q: What is the Pythagorean identity that we use in the proof?

A: The Pythagorean identity that we use in the proof is $\operatorname{Sin}^2 \theta + \operatorname{Cos}^2 \theta = 1$.

Q: How do we eliminate the square root in the proof?

A: We eliminate the square root by squaring both sides of the equation.

Q: What is the final expression for $\operatorname{Cos} \theta$ that we obtain in the proof?

A: The final expression for $\operatorname{Cos} \theta$ that we obtain in the proof is $\operatorname{Cos} \theta = \frac{p^2 - 1}{p^2 + 1}$.

Q: What is the significance of this proof?

A: This proof shows that if $\operatorname{Cosec} \theta + \operatorname{Cot} \theta = p$, then we can express $\operatorname{Cos} \theta$ in terms of $p$.

Q: What are some potential applications of this proof?

A: This proof may have applications in trigonometry, calculus, and other areas of mathematics where trigonometric functions are used.

Q: How can we use this proof to solve problems?

A: We can use this proof to solve problems that involve trigonometric functions and their relationships.

Q: What are some potential extensions of this proof?

A: Some potential extensions of this proof include exploring the relationships between other trigonometric functions and using this proof to derive new identities.

Q: What are some potential challenges in using this proof?

A: Some potential challenges in using this proof include understanding the underlying trigonometric identities and being able to manipulate the equations effectively.

Q: How can we verify the correctness of this proof?

A: We can verify the correctness of this proof by checking the steps and ensuring that they are logically consistent.

Q: What are some potential limitations of this proof?

A: Some potential limitations of this proof include its reliance on specific trigonometric identities and its applicability to specific types of problems.

Q: How can we use this proof to explore new ideas?

A: We can use this proof to explore new ideas by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential areas of research related to this proof?

A: Some potential areas of research related to this proof include exploring the relationships between trigonometric functions and their applications in various fields.

Q: How can we use this proof to improve our understanding of trigonometry?

A: We can use this proof to improve our understanding of trigonometry by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential benefits of using this proof?

A: Some potential benefits of using this proof include gaining a deeper understanding of trigonometric functions and their relationships, and being able to apply this knowledge to solve problems.

Q: How can we use this proof to improve our problem-solving skills?

A: We can use this proof to improve our problem-solving skills by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential challenges in applying this proof to real-world problems?

A: Some potential challenges in applying this proof to real-world problems include understanding the underlying trigonometric identities and being able to manipulate the equations effectively.

Q: How can we use this proof to explore new areas of mathematics?

A: We can use this proof to explore new areas of mathematics by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential areas of application for this proof?

A: Some potential areas of application for this proof include trigonometry, calculus, and other areas of mathematics where trigonometric functions are used.

Q: How can we use this proof to improve our understanding of mathematical relationships?

A: We can use this proof to improve our understanding of mathematical relationships by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential benefits of using this proof in education?

A: Some potential benefits of using this proof in education include providing students with a deeper understanding of trigonometric functions and their relationships, and helping them to develop problem-solving skills.

Q: How can we use this proof to improve our understanding of mathematical concepts?

A: We can use this proof to improve our understanding of mathematical concepts by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential areas of research related to this proof in education?

A: Some potential areas of research related to this proof in education include exploring the effectiveness of using this proof in teaching trigonometry and other mathematical concepts.

Q: How can we use this proof to improve our understanding of mathematical relationships in education?

A: We can use this proof to improve our understanding of mathematical relationships in education by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential benefits of using this proof in education?

A: Some potential benefits of using this proof in education include providing students with a deeper understanding of trigonometric functions and their relationships, and helping them to develop problem-solving skills.

Q: How can we use this proof to improve our understanding of mathematical concepts in education?

A: We can use this proof to improve our understanding of mathematical concepts in education by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential areas of research related to this proof in education?

A: Some potential areas of research related to this proof in education include exploring the effectiveness of using this proof in teaching trigonometry and other mathematical concepts.

Q: How can we use this proof to improve our understanding of mathematical relationships in education?

A: We can use this proof to improve our understanding of mathematical relationships in education by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential benefits of using this proof in education?

A: Some potential benefits of using this proof in education include providing students with a deeper understanding of trigonometric functions and their relationships, and helping them to develop problem-solving skills.

Q: How can we use this proof to improve our understanding of mathematical concepts in education?

A: We can use this proof to improve our understanding of mathematical concepts in education by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential areas of research related to this proof in education?

A: Some potential areas of research related to this proof in education include exploring the effectiveness of using this proof in teaching trigonometry and other mathematical concepts.

Q: How can we use this proof to improve our understanding of mathematical relationships in education?

A: We can use this proof to improve our understanding of mathematical relationships in education by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential benefits of using this proof in education?

A: Some potential benefits of using this proof in education include providing students with a deeper understanding of trigonometric functions and their relationships, and helping them to develop problem-solving skills.

Q: How can we use this proof to improve our understanding of mathematical concepts in education?

A: We can use this proof to improve our understanding of mathematical concepts in education by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential areas of research related to this proof in education?

A: Some potential areas of research related to this proof in education include exploring the effectiveness of using this proof in teaching trigonometry and other mathematical concepts.

Q: How can we use this proof to improve our understanding of mathematical relationships in education?

A: We can use this proof to improve our understanding of mathematical relationships in education by applying it to different types of problems and seeing how it can be used to derive new identities and relationships.

Q: What are some potential benefits of using this proof in education?

A: Some potential benefits of using this proof in education include providing students with a deeper understanding of trigonometric functions and their relationships, and helping them to develop problem-solving skills.

Q: How can we use this