Prove the identity. (1- sin'x)c CSCX= COSX cotx
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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Can someone help me prove the identity of the statement? It has to be step by step with the rule included.
![### Proving Trigonometric Identities: An Example
#### Example: Verify the identity
\[ (1 - \sin^2 x) \csc x = \cos x \cot x \]
To prove this identity, we will show each step and the rules applied in transforming the left-hand side (LHS) to make it equal to the right-hand side (RHS).
#### Steps:
1. **Rewrite the given identity:**
\[ (1 - \sin^2 x) \csc x \]
2. **Apply the Pythagorean Identity:**
Recall that \(\sin^2 x + \cos^2 x = 1\), so:
\[ 1 - \sin^2 x = \cos^2 x \]
Substitute this into our equation:
\[ \cos^2 x \csc x \]
3. **Rewrite \(\csc x\):**
Remember that \(\csc x = \frac{1}{\sin x}\), so:
\[ \cos^2 x \cdot \frac{1}{\sin x} \]
4. **Simplify:**
Separate the terms:
\[ \cos x \cdot \left(\cos x \cdot \frac{1}{\sin x}\right) \]
Since \(\frac{\cos x}{\sin x} = \cot x\):
\[ \cos x \cdot \cot x \]
Thus, we have shown that:
\[ (1 - \sin^2 x) \csc x = \cos x \cot x \]
### Key Points to Remember:
- Use the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\)
- Remember reciprocal identities: \(\csc x = \frac{1}{\sin x}\)
- Simplify step by step, ensuring to apply algebraic transformations correctly
Ensuring each statement is based on a recognized rule or identity helps in systematically proving trigonometric identities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F754805ce-a10b-4c4c-8d6b-69e24af64694%2F5a9a5fcc-4e7c-404e-bad8-d93b585777b4%2Fmejnkt.jpeg&w=3840&q=75)
Transcribed Image Text:### Proving Trigonometric Identities: An Example
#### Example: Verify the identity
\[ (1 - \sin^2 x) \csc x = \cos x \cot x \]
To prove this identity, we will show each step and the rules applied in transforming the left-hand side (LHS) to make it equal to the right-hand side (RHS).
#### Steps:
1. **Rewrite the given identity:**
\[ (1 - \sin^2 x) \csc x \]
2. **Apply the Pythagorean Identity:**
Recall that \(\sin^2 x + \cos^2 x = 1\), so:
\[ 1 - \sin^2 x = \cos^2 x \]
Substitute this into our equation:
\[ \cos^2 x \csc x \]
3. **Rewrite \(\csc x\):**
Remember that \(\csc x = \frac{1}{\sin x}\), so:
\[ \cos^2 x \cdot \frac{1}{\sin x} \]
4. **Simplify:**
Separate the terms:
\[ \cos x \cdot \left(\cos x \cdot \frac{1}{\sin x}\right) \]
Since \(\frac{\cos x}{\sin x} = \cot x\):
\[ \cos x \cdot \cot x \]
Thus, we have shown that:
\[ (1 - \sin^2 x) \csc x = \cos x \cot x \]
### Key Points to Remember:
- Use the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\)
- Remember reciprocal identities: \(\csc x = \frac{1}{\sin x}\)
- Simplify step by step, ensuring to apply algebraic transformations correctly
Ensuring each statement is based on a recognized rule or identity helps in systematically proving trigonometric identities.
Transcribed Image Text:### Select the Rule
Please choose one of the following mathematical rules to explore:
- **Algebra**
- **Reciprocal**
- **Quotient**
- **Pythagorean**
- **Odd Even**
Click on the radio button next to the rule you wish to learn more about. Each rule comes with detailed explanations, examples, and exercises to enhance your understanding.
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