Simplify The Expression:$ \frac{1}{\sin X} - \sin X }$Choose The Correct Simplification A. { \cos X$ $B. { \tan X$}$

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Introduction

Simplifying trigonometric expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying the expression $\frac{1}{\sin x} - \sin x$. This expression involves the sine function and its reciprocal, and it can be simplified using various trigonometric identities.

Understanding the Expression

The given expression is $\frac{1}{\sin x} - \sin x$. To simplify this expression, we need to understand the properties of the sine function and its reciprocal. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The reciprocal of the sine function is the cosecant function, which is defined as the ratio of the length of the hypotenuse to the length of the opposite side.

Using Trigonometric Identities

To simplify the expression, we can use various trigonometric identities. One of the most commonly used identities is the Pythagorean identity, which states that $\sin^2 x + \cos^2 x = 1$. We can also use the identity $\csc x = \frac{1}{\sin x}$, which is the reciprocal of the sine function.

Simplifying the Expression

Using the Pythagorean identity, we can rewrite the expression as follows:

$$\frac{1}{\sin x} - \sin x = \frac{1 - \sin^2 x}{\sin x}$$

Now, we can use the identity $\sin^2 x + \cos^2 x = 1$ to simplify the expression further:

$$\frac{1 - \sin^2 x}{\sin x} = \frac{\cos^2 x}{\sin x}$$

Using the Reciprocal Identity

We can use the reciprocal identity $\csc x = \frac{1}{\sin x}$ to rewrite the expression as follows:

$$\frac{\cos^2 x}{\sin x} = \cos x \cdot \csc x$$

Simplifying the Expression Further

Using the identity $\csc x = \frac{1}{\sin x}$, we can rewrite the expression as follows:

$$\cos x \cdot \csc x = \cos x \cdot \frac{1}{\sin x}$$

Final Simplification

The final simplification of the expression is:

$$\cos x \cdot \frac{1}{\sin x} = \frac{\cos x}{\sin x}$$

Conclusion

In conclusion, the correct simplification of the expression $\frac{1}{\sin x} - \sin x$ is $\frac{\cos x}{\sin x}$. This expression can be rewritten as $\cot x$, which is the reciprocal of the tangent function.

Final Answer

The final answer is $\boxed{\cot x}$.

Discussion

The discussion category for this article is mathematics. The article focuses on simplifying the expression $\frac{1}{\sin x} - \sin x$ using various trigonometric identities. The final simplification of the expression is $\frac{\cos x}{\sin x}$, which can be rewritten as $\cot x$. This article provides a step-by-step guide on how to simplify the expression and provides a deep understanding of the underlying concepts.

Related Articles

• Simplifying Trigonometric Expressions
• Trigonometric Identities
• Pythagorean Identity
• Reciprocal Identity

Keywords

• Simplifying trigonometric expressions
• Trigonometric identities
• Pythagorean identity
• Reciprocal identity
• Cotangent function
• Tangent function
• Sine function
• Cosine function
  

Introduction

In our previous article, we discussed how to simplify the expression $\frac{1}{\sin x} - \sin x$. We used various trigonometric identities to arrive at the final simplification of $\frac{\cos x}{\sin x}$, which can be rewritten as $\cot x$. In this article, we will answer some frequently asked questions related to simplifying trigonometric expressions.

Q1: What is the Pythagorean identity?

A1: The Pythagorean identity is a fundamental trigonometric identity that states $\sin^2 x + \cos^2 x = 1$. This identity is used to simplify trigonometric expressions and is a crucial concept in trigonometry.

Q2: How do I simplify the expression $\frac{1}{\sin x} - \sin x$?

A2: To simplify the expression $\frac{1}{\sin x} - \sin x$, we can use the Pythagorean identity and the reciprocal identity. We can rewrite the expression as $\frac{1 - \sin^2 x}{\sin x}$ and then simplify it further using the Pythagorean identity.

Q3: What is the reciprocal identity?

A3: The reciprocal identity is a trigonometric identity that states $\csc x = \frac{1}{\sin x}$ and $\sec x = \frac{1}{\cos x}$. This identity is used to simplify trigonometric expressions and is a crucial concept in trigonometry.

Q4: How do I simplify the expression $\cos x \cdot \csc x$?

A4: To simplify the expression $\cos x \cdot \csc x$, we can use the reciprocal identity. We can rewrite the expression as $\cos x \cdot \frac{1}{\sin x}$ and then simplify it further.

Q5: What is the final simplification of the expression $\frac{1}{\sin x} - \sin x$?

A5: The final simplification of the expression $\frac{1}{\sin x} - \sin x$ is $\frac{\cos x}{\sin x}$, which can be rewritten as $\cot x$.

Q6: How do I use trigonometric identities to simplify expressions?

A6: To use trigonometric identities to simplify expressions, you need to identify the relevant identities and apply them step by step. You can use the Pythagorean identity, the reciprocal identity, and other trigonometric identities to simplify expressions.

Q7: What are some common trigonometric identities?

A7: Some common trigonometric identities include the Pythagorean identity, the reciprocal identity, and the sum and difference identities. These identities are used to simplify trigonometric expressions and are a crucial concept in trigonometry.

Q8: How do I apply the Pythagorean identity to simplify expressions?

A8: To apply the Pythagorean identity to simplify expressions, you need to identify the relevant terms and rewrite them using the Pythagorean identity. For example, you can rewrite $\sin^2 x + \cos^2 x$ as $1$.

Q9: What is the cotangent function?

A9: The cotangent function is the reciprocal of the tangent function. It is defined as $\cot x = \frac{\cos x}{\sin x}$.

Q10: How do I use the cotangent function to simplify expressions?

A10: To use the cotangent function to simplify expressions, you need to identify the relevant terms and rewrite them using the cotangent function. For example, you can rewrite $\frac{\cos x}{\sin x}$ as $\cot x$.

Conclusion

In conclusion, simplifying trigonometric expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. By using trigonometric identities, such as the Pythagorean identity and the reciprocal identity, we can simplify expressions and arrive at the final simplification. We hope that this Q&A article has provided a helpful guide on how to simplify trigonometric expressions.

Related Articles

• Simplifying Trigonometric Expressions
• Trigonometric Identities
• Pythagorean Identity
• Reciprocal Identity
• Cotangent Function

Keywords

• Simplifying trigonometric expressions
• Trigonometric identities
• Pythagorean identity
• Reciprocal identity
• Cotangent function
• Tangent function
• Sine function
• Cosine function