Melanie Simplified The Expression $\left(\frac{1}{\sec (x)}\right$\] As Shown Below.Step 1: $\frac{\left(\frac{\cos (x)}{\sin (x)}\right)}{(\cos (x))}$Step 2: $\quad \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)}$Step 3:

Introduction

Trigonometric expressions can be complex and challenging to simplify, but with the right approach, they can be broken down into manageable parts. In this article, we will explore how to simplify a given trigonometric expression using step-by-step instructions. We will focus on the expression $\left(\frac{1}{\sec (x)}\right)$ and simplify it using trigonometric identities.

Step 1: Simplifying the Expression

The given expression is $\left(\frac{1}{\sec (x)}\right)$. To simplify this expression, we can start by rewriting it in terms of sine and cosine.

$\frac{\left(\frac{\cos (x)}{\sin (x)}\right)}{(\cos (x))}$

In this step, we have rewritten the expression using the reciprocal identity of secant, which is $\sec (x) = \frac{1}{\cos (x)}$. We have also used the fact that $\frac{\cos (x)}{\sin (x)} = \tan (x)$.

Step 2: Simplifying Further

Now that we have rewritten the expression, we can simplify it further by canceling out the common factors.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)}$

In this step, we have canceled out the $\cos (x)$ terms, leaving us with $\frac{1}{\sin (x)}$.

Step 3: Final Simplification

The final simplified expression is $\frac{1}{\sin (x)}$. This is the simplified form of the original expression $\left(\frac{1}{\sec (x)}\right)$.

Discussion

In this article, we have simplified a given trigonometric expression using step-by-step instructions. We have used trigonometric identities to rewrite the expression and simplify it further. The final simplified expression is $\frac{1}{\sin (x)}$.

Conclusion

Simplifying trigonometric expressions can be challenging, but with the right approach, they can be broken down into manageable parts. By using trigonometric identities and simplifying step-by-step, we can arrive at the final simplified expression. In this article, we have demonstrated how to simplify a given trigonometric expression using step-by-step instructions.

Common Trigonometric Identities

Here are some common trigonometric identities that are useful for simplifying trigonometric expressions:

• $\sec (x) = \frac{1}{\cos (x)}$
• $\csc (x) = \frac{1}{\sin (x)}$
• $\tan (x) = \frac{\sin (x)}{\cos (x)}$
• $\cot (x) = \frac{\cos (x)}{\sin (x)}$

Tips for Simplifying Trigonometric Expressions

Here are some tips for simplifying trigonometric expressions:

• Use trigonometric identities to rewrite the expression.
• Simplify step-by-step, canceling out common factors.
• Use the reciprocal identities of secant and cosecant.
• Use the fact that $\tan (x) = \frac{\sin (x)}{\cos (x)}$ and $\cot (x) = \frac{\cos (x)}{\sin (x)}$.

Practice Problems

Here are some practice problems to help you practice simplifying trigonometric expressions:

• Simplify the expression $\left(\frac{1}{\csc (x)}\right)$.
• Simplify the expression $\left(\frac{\tan (x)}{\sec (x)}\right)$.
• Simplify the expression $\left(\frac{\cot (x)}{\csc (x)}\right)$.

Answer Key

Here are the answers to the practice problems:

• $\sin (x)$
• $\cos (x)$
• $\sin (x)$
  Frequently Asked Questions: Simplifying Trigonometric Expressions ====================================================================

Q: What is the reciprocal identity of secant?

A: The reciprocal identity of secant is $\sec (x) = \frac{1}{\cos (x)}$.

Q: How do I simplify a trigonometric expression using step-by-step instructions?

A: To simplify a trigonometric expression using step-by-step instructions, follow these steps:

1. Rewrite the expression using trigonometric identities.
2. Simplify step-by-step, canceling out common factors.
3. Use the reciprocal identities of secant and cosecant.
4. Use the fact that $\tan (x) = \frac{\sin (x)}{\cos (x)}$ and $\cot (x) = \frac{\cos (x)}{\sin (x)}$.

Q: What are some common trigonometric identities that I should know?

A: Here are some common trigonometric identities that are useful for simplifying trigonometric expressions:

• $\sec (x) = \frac{1}{\cos (x)}$
• $\csc (x) = \frac{1}{\sin (x)}$
• $\tan (x) = \frac{\sin (x)}{\cos (x)}$
• $\cot (x) = \frac{\cos (x)}{\sin (x)}$

Q: How do I simplify the expression $\left(\frac{1}{\csc (x)}\right)$?

A: To simplify the expression $\left(\frac{1}{\csc (x)}\right)$, we can use the reciprocal identity of cosecant, which is $\csc (x) = \frac{1}{\sin (x)}$. Therefore, the simplified expression is $\sin (x)$.

Q: How do I simplify the expression $\left(\frac{\tan (x)}{\sec (x)}\right)$?

A: To simplify the expression $\left(\frac{\tan (x)}{\sec (x)}\right)$, we can use the fact that $\tan (x) = \frac{\sin (x)}{\cos (x)}$ and $\sec (x) = \frac{1}{\cos (x)}$. Therefore, the simplified expression is $\sin (x)$.

Q: How do I simplify the expression $\left(\frac{\cot (x)}{\csc (x)}\right)$?

A: To simplify the expression $\left(\frac{\cot (x)}{\csc (x)}\right)$, we can use the fact that $\cot (x) = \frac{\cos (x)}{\sin (x)}$ and $\csc (x) = \frac{1}{\sin (x)}$. Therefore, the simplified expression is $\cos (x)$.

Q: What are some tips for simplifying trigonometric expressions?

A: Here are some tips for simplifying trigonometric expressions:

• Use trigonometric identities to rewrite the expression.
• Simplify step-by-step, canceling out common factors.
• Use the reciprocal identities of secant and cosecant.
• Use the fact that $\tan (x) = \frac{\sin (x)}{\cos (x)}$ and $\cot (x) = \frac{\cos (x)}{\sin (x)}$.

Q: How do I practice simplifying trigonometric expressions?

A: To practice simplifying trigonometric expressions, try the following:

• Simplify the expression $\left(\frac{1}{\csc (x)}\right)$.
• Simplify the expression $\left(\frac{\tan (x)}{\sec (x)}\right)$.
• Simplify the expression $\left(\frac{\cot (x)}{\csc (x)}\right)$.

Conclusion

Simplifying trigonometric expressions can be challenging, but with the right approach, they can be broken down into manageable parts. By using trigonometric identities and simplifying step-by-step, we can arrive at the final simplified expression. In this article, we have demonstrated how to simplify a given trigonometric expression using step-by-step instructions and provided some tips and practice problems to help you practice simplifying trigonometric expressions.