Prove the identity. TC sin +x+ 2. cotx sin (n-x)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
How can I prove the identity step by step?
![### Proving Trigonometric Identities: Practice Exercise
To strengthen your understanding of trigonometric identities and equations, let's prove the given identity:
\[ \frac{\sin\left( \frac{\pi}{2} + x \right)}{\sin(\pi - x)} = \cot{x} \]
The task is to demonstrate that the left-hand side of the equation is equal to the right-hand side, using established trigonometric rules and identities.
#### Steps to Prove the Identity
1. **Rewrite Trigonometric Functions Using Known Identities:**
Utilize trigonometric identities to simplify both the numerator and the denominator.
2. **Simplify Expressions:**
Simplify the resulting expressions to show that they equal \(\cot{x}\).
### Detailed Analysis:
- **Initial Setup:**
The given identity to be proved is:
\[
\frac{\sin\left(\frac{\pi}{2} + x\right)}{\sin(\pi - x)} = \cot{x}
\]
- **Applying Trigonometric Identities:**
Recall the following identities:
\[
\sin\left( \frac{\pi}{2} + x \right) = \cos{x}
\]
\[
\sin(\pi - x) = \sin{x}
\]
- **Substitute These Identities:**
Substituting the above identities into the given expression:
\[
\frac{\cos{x}}{\sin{x}}
\]
- **Simplify to Proof:**
Recognize that \(\frac{\cos{x}}{\sin{x}} = \cot{x}\).
Thus, the given identity is proven.
\[ \frac{\sin\left( \frac{\pi}{2} + x \right)}{\sin(\pi - x)} = \cot{x} \]
### Explanation of the Diagram below:
A section of the screen captures a built-in tool for validating the steps taken to prove the identity. Specifically, it includes a box for entering each step of the proof and the corresponding rules applied.
1. **Statement Box:**
It reads:
\[
\frac{\sin\left( \frac{\pi}{2} + x \right)}{\sin(\pi - x)} =
\]
This is where you would start inputting steps to show that the identity holds.
2. **Validate Button:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F754805ce-a10b-4c4c-8d6b-69e24af64694%2Fdca70272-a20a-4cc7-8ec8-84f2f98ff342%2Ff7139x8_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Proving Trigonometric Identities: Practice Exercise
To strengthen your understanding of trigonometric identities and equations, let's prove the given identity:
\[ \frac{\sin\left( \frac{\pi}{2} + x \right)}{\sin(\pi - x)} = \cot{x} \]
The task is to demonstrate that the left-hand side of the equation is equal to the right-hand side, using established trigonometric rules and identities.
#### Steps to Prove the Identity
1. **Rewrite Trigonometric Functions Using Known Identities:**
Utilize trigonometric identities to simplify both the numerator and the denominator.
2. **Simplify Expressions:**
Simplify the resulting expressions to show that they equal \(\cot{x}\).
### Detailed Analysis:
- **Initial Setup:**
The given identity to be proved is:
\[
\frac{\sin\left(\frac{\pi}{2} + x\right)}{\sin(\pi - x)} = \cot{x}
\]
- **Applying Trigonometric Identities:**
Recall the following identities:
\[
\sin\left( \frac{\pi}{2} + x \right) = \cos{x}
\]
\[
\sin(\pi - x) = \sin{x}
\]
- **Substitute These Identities:**
Substituting the above identities into the given expression:
\[
\frac{\cos{x}}{\sin{x}}
\]
- **Simplify to Proof:**
Recognize that \(\frac{\cos{x}}{\sin{x}} = \cot{x}\).
Thus, the given identity is proven.
\[ \frac{\sin\left( \frac{\pi}{2} + x \right)}{\sin(\pi - x)} = \cot{x} \]
### Explanation of the Diagram below:
A section of the screen captures a built-in tool for validating the steps taken to prove the identity. Specifically, it includes a box for entering each step of the proof and the corresponding rules applied.
1. **Statement Box:**
It reads:
\[
\frac{\sin\left( \frac{\pi}{2} + x \right)}{\sin(\pi - x)} =
\]
This is where you would start inputting steps to show that the identity holds.
2. **Validate Button:
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