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 RulesThis 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.---### SummaryIn 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. ### 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.

Some common trignometry identies are :Sinπ2+x = Cos xCosπ2-x = +Sin xCosπ2+x = -Sin xtanπ2+x = -Cot…

Answered: Prove the identity. COS x+ 2 = tanx C.…

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Trigonometry

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.

### 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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