Prove the identity. COS x+ 2 = tanx C. cos (+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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Question
How can I prove the identity step by step with the rule?
![### Trigonometric Identity Proof
#### Prove the identity:
\[
\frac{\cos\left(\frac{\pi}{2} + x\right)}{\cos(\pi + x)} = \tan x
\]
Note that each statement must be based on a rule that should be listed to the right of the statement.
---
#### Statements and Rules
This section can be used to write down each step taken to prove the above identity along with the rule used for that step.
**Statement** | **Rule**
:------------: | :---------:
\(\cos\left(\frac{\pi}{2} + x\right)\) | Use \(\cos(\frac{\pi}{2} + x) = -\sin(x)\)
\(\cos(\pi + x)\) | Use \(\cos(\pi + x) = -\cos(x)\)
\(\frac{-\sin(x)}{-\cos(x)}\) | Substitute the expressions obtained above
\(\frac{\sin(x)}{\cos(x)}\) | Simplify by canceling out negative signs
\(\tan(x)\) | Use the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)
By following these steps, you will successfully prove the given trigonometric identity.
---
### Summary
In conclusion, using trigonometric identities and manipulation of trigonometric functions, the given identity \(\frac{\cos\left(\frac{\pi}{2} + x\right)}{\cos(\pi + x)} = \tan x\) has been proven. Each step adheres to commonly known identities and simplification processes in trigonometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F754805ce-a10b-4c4c-8d6b-69e24af64694%2F50d22962-004c-47c7-a95c-26d223716467%2Fvgavfxr.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Identity Proof
#### Prove the identity:
\[
\frac{\cos\left(\frac{\pi}{2} + x\right)}{\cos(\pi + x)} = \tan x
\]
Note that each statement must be based on a rule that should be listed to the right of the statement.
---
#### Statements and Rules
This section can be used to write down each step taken to prove the above identity along with the rule used for that step.
**Statement** | **Rule**
:------------: | :---------:
\(\cos\left(\frac{\pi}{2} + x\right)\) | Use \(\cos(\frac{\pi}{2} + x) = -\sin(x)\)
\(\cos(\pi + x)\) | Use \(\cos(\pi + x) = -\cos(x)\)
\(\frac{-\sin(x)}{-\cos(x)}\) | Substitute the expressions obtained above
\(\frac{\sin(x)}{\cos(x)}\) | Simplify by canceling out negative signs
\(\tan(x)\) | Use the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)
By following these steps, you will successfully prove the given trigonometric identity.
---
### Summary
In conclusion, using trigonometric identities and manipulation of trigonometric functions, the given identity \(\frac{\cos\left(\frac{\pi}{2} + x\right)}{\cos(\pi + x)} = \tan x\) has been proven. Each step adheres to commonly known identities and simplification processes in trigonometry.
Transcribed Image Text:### Select the Rule
**Please select the mathematical rule you would like to learn more about:**
- **Algebra**
- **Reciprocal**
- **Quotient**
- **Pythagorean**
- **Odd Even**
Simply click on the circle next to the rule to begin exploring detailed explanations and examples.
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