Simplify The Expression:${ \frac{\sin^2 X - \cos^2 X}{\cos X[\sin(180^\circ - X) - \cos X]} - 1 = \tan X }$

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Introduction

Trigonometric identities are a fundamental concept in mathematics, and they play a crucial role in solving various problems in physics, engineering, and other fields. One of the most common trigonometric identities is the Pythagorean identity, which states that sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1. However, in this article, we will focus on simplifying a more complex expression involving trigonometric functions. The given expression is:

sin⁡2x−cos⁡2xcos⁡x[sin⁡(180∘−x)−cos⁡x]−1=tan⁡x\frac{\sin^2 x - \cos^2 x}{\cos x[\sin(180^\circ - x) - \cos x]} - 1 = \tan x

Our goal is to simplify this expression and understand its underlying structure.

Understanding the Given Expression

The given expression involves several trigonometric functions, including sine, cosine, and tangent. To simplify this expression, we need to understand the properties and relationships between these functions. Let's start by analyzing the numerator of the expression:

sin⁡2x−cos⁡2x\sin^2 x - \cos^2 x

This expression can be rewritten using the Pythagorean identity as:

sin⁡2x−cos⁡2x=(sin⁡x+cos⁡x)(sin⁡x−cos⁡x)\sin^2 x - \cos^2 x = (\sin x + \cos x)(\sin x - \cos x)

This is a difference of squares, which can be factored into the product of two binomials.

Simplifying the Numerator

Now, let's focus on simplifying the numerator of the expression. We can start by factoring the difference of squares:

sin⁡2x−cos⁡2x=(sin⁡x+cos⁡x)(sin⁡x−cos⁡x)\sin^2 x - \cos^2 x = (\sin x + \cos x)(\sin x - \cos x)

This can be further simplified by recognizing that sin⁡x−cos⁡x\sin x - \cos x is the negative of cos⁡x−sin⁡x\cos x - \sin x. Therefore, we can rewrite the numerator as:

sin⁡2x−cos⁡2x=(sin⁡x+cos⁡x)(cos⁡x−sin⁡x)\sin^2 x - \cos^2 x = (\sin x + \cos x)(\cos x - \sin x)

Simplifying the Denominator

Now, let's focus on simplifying the denominator of the expression. We can start by analyzing the term sin⁡(180∘−x)\sin(180^\circ - x). Using the identity sin⁡(180∘−x)=sin⁡x\sin(180^\circ - x) = \sin x, we can rewrite the denominator as:

cos⁡x[sin⁡(180∘−x)−cos⁡x]=cos⁡x[sin⁡x−cos⁡x]\cos x[\sin(180^\circ - x) - \cos x] = \cos x[\sin x - \cos x]

This can be further simplified by recognizing that sin⁡x−cos⁡x\sin x - \cos x is the negative of cos⁡x−sin⁡x\cos x - \sin x. Therefore, we can rewrite the denominator as:

cos⁡x[sin⁡x−cos⁡x]=cos⁡x(cos⁡x−sin⁡x)\cos x[\sin x - \cos x] = \cos x(\cos x - \sin x)

Combining the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to simplify the expression:

sin⁡2x−cos⁡2xcos⁡x[sin⁡(180∘−x)−cos⁡x]−1=(sin⁡x+cos⁡x)(cos⁡x−sin⁡x)cos⁡x(cos⁡x−sin⁡x)−1\frac{\sin^2 x - \cos^2 x}{\cos x[\sin(180^\circ - x) - \cos x]} - 1 = \frac{(\sin x + \cos x)(\cos x - \sin x)}{\cos x(\cos x - \sin x)} - 1

Canceling Common Factors

We can simplify this expression further by canceling common factors. The numerator and denominator both contain the factor (cos⁡x−sin⁡x)(\cos x - \sin x). Therefore, we can cancel this factor to obtain:

(sin⁡x+cos⁡x)(cos⁡x−sin⁡x)cos⁡x(cos⁡x−sin⁡x)−1=sin⁡x+cos⁡xcos⁡x−1\frac{(\sin x + \cos x)(\cos x - \sin x)}{\cos x(\cos x - \sin x)} - 1 = \frac{\sin x + \cos x}{\cos x} - 1

Simplifying the Expression

Now that we have simplified the expression, we can further simplify it by recognizing that sin⁡x+cos⁡xcos⁡x\frac{\sin x + \cos x}{\cos x} is the negative of tan⁡x\tan x. Therefore, we can rewrite the expression as:

sin⁡x+cos⁡xcos⁡x−1=−tan⁡x−1\frac{\sin x + \cos x}{\cos x} - 1 = -\tan x - 1

Conclusion

In this article, we simplified a complex expression involving trigonometric functions. We started by analyzing the numerator and denominator of the expression and simplified them separately. We then combined the simplified numerator and denominator to obtain a simplified expression. Finally, we recognized that the simplified expression is equivalent to −tan⁡x−1-\tan x - 1. This demonstrates the importance of understanding the properties and relationships between trigonometric functions in simplifying complex expressions.

Final Answer

The final answer is −tan⁡x−1\boxed{-\tan x - 1}.

Additional Resources

For more information on trigonometric identities and simplifying complex expressions, please refer to the following resources:

Introduction

In our previous article, we simplified a complex expression involving trigonometric functions. We started by analyzing the numerator and denominator of the expression and simplified them separately. We then combined the simplified numerator and denominator to obtain a simplified expression. Finally, we recognized that the simplified expression is equivalent to −tan⁡x−1-\tan x - 1. In this article, we will answer some common questions related to the simplification of the expression.

Q&A

Q: What is the main concept behind simplifying the expression?

A: The main concept behind simplifying the expression is to understand the properties and relationships between trigonometric functions. We need to recognize that the expression can be simplified by factoring the numerator and denominator, and then combining the simplified terms.

Q: How do we simplify the numerator of the expression?

A: We can simplify the numerator of the expression by factoring the difference of squares. The numerator can be rewritten as (sin⁡x+cos⁡x)(sin⁡x−cos⁡x)(\sin x + \cos x)(\sin x - \cos x), which can be further simplified by recognizing that sin⁡x−cos⁡x\sin x - \cos x is the negative of cos⁡x−sin⁡x\cos x - \sin x.

Q: How do we simplify the denominator of the expression?

A: We can simplify the denominator of the expression by recognizing that sin⁡(180∘−x)=sin⁡x\sin(180^\circ - x) = \sin x. Therefore, the denominator can be rewritten as cos⁡x(cos⁡x−sin⁡x)\cos x(\cos x - \sin x).

Q: How do we combine the simplified numerator and denominator?

A: We can combine the simplified numerator and denominator by canceling common factors. The numerator and denominator both contain the factor (cos⁡x−sin⁡x)(\cos x - \sin x). Therefore, we can cancel this factor to obtain the simplified expression.

Q: What is the final simplified expression?

A: The final simplified expression is −tan⁡x−1-\tan x - 1.

Q: What are some common mistakes to avoid when simplifying the expression?

A: Some common mistakes to avoid when simplifying the expression include:

  • Not recognizing the difference of squares in the numerator
  • Not recognizing the identity sin⁡(180∘−x)=sin⁡x\sin(180^\circ - x) = \sin x in the denominator
  • Not canceling common factors in the numerator and denominator
  • Not recognizing that the simplified expression is equivalent to −tan⁡x−1-\tan x - 1

Q: What are some real-world applications of simplifying the expression?

A: Simplifying the expression has several real-world applications, including:

  • Solving trigonometric equations
  • Simplifying complex trigonometric expressions
  • Understanding the properties and relationships between trigonometric functions
  • Applying trigonometric identities to solve problems in physics, engineering, and other fields

Conclusion

In this article, we answered some common questions related to the simplification of the expression. We discussed the main concept behind simplifying the expression, how to simplify the numerator and denominator, how to combine the simplified terms, and what are some common mistakes to avoid. We also discussed some real-world applications of simplifying the expression. By understanding the properties and relationships between trigonometric functions, we can simplify complex expressions and apply trigonometric identities to solve problems in various fields.

Final Answer

The final answer is −tan⁡x−1\boxed{-\tan x - 1}.

Additional Resources

For more information on trigonometric identities and simplifying complex expressions, please refer to the following resources: