Prove the identity. COS 2 - tanx cos (T+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
Could someone help me prove the identity step by step? I am confused.
![**Proving the Trigonometric Identity**
Prove the identity:
\[
\frac{\cos \left(\frac{\pi}{2} + x\right)}{\cos \left(\pi + x\right)} = \tan x
\]
Note that each Statement must be based on a Rule, represented to the right of the Statement.
---
### Detailed Steps:
1. **Start with the given equation:**
\[
\frac{\cos \left(\frac{\pi}{2} + x\right)}{\cos (\pi + x)} \stackrel{?}{=} \tan x
\]
2. **Use trigonometric identities for cosine addition formulas:**
\[
\cos \left(\frac{\pi}{2} + x\right) = -\sin x \quad \text{(Trigonometric identity: }\cos(A + B) = \cos A \cos B - \sin A \sin B)
\]
\[
\cos (\pi + x) = -\cos x \quad \text{(Trigonometric identity: }\cos(A + B) = -\cos A \text{ when } B = \pi)
\]
3. **Substitute these values into the original equation:**
\[
\frac{-\sin x}{-\cos x}
\]
4. **Simplify the expression:**
\[
\frac{-\sin x}{-\cos x} = \frac{\sin x}{\cos x}
\]
5. **Recognize the resulting trigonometric ratio:**
\[
\frac{\sin x}{\cos x} = \tan x
\]
Thus, we have shown that:
\[
\frac{\cos \left(\frac{\pi}{2} + x\right)}{\cos (\pi + x)} = \tan x
\]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F754805ce-a10b-4c4c-8d6b-69e24af64694%2Fb0100fa1-4c71-4499-9d1e-203e45bb5b81%2F1u3hlwg.jpeg&w=3840&q=75)
Transcribed Image Text:**Proving the Trigonometric Identity**
Prove the identity:
\[
\frac{\cos \left(\frac{\pi}{2} + x\right)}{\cos \left(\pi + x\right)} = \tan x
\]
Note that each Statement must be based on a Rule, represented to the right of the Statement.
---
### Detailed Steps:
1. **Start with the given equation:**
\[
\frac{\cos \left(\frac{\pi}{2} + x\right)}{\cos (\pi + x)} \stackrel{?}{=} \tan x
\]
2. **Use trigonometric identities for cosine addition formulas:**
\[
\cos \left(\frac{\pi}{2} + x\right) = -\sin x \quad \text{(Trigonometric identity: }\cos(A + B) = \cos A \cos B - \sin A \sin B)
\]
\[
\cos (\pi + x) = -\cos x \quad \text{(Trigonometric identity: }\cos(A + B) = -\cos A \text{ when } B = \pi)
\]
3. **Substitute these values into the original equation:**
\[
\frac{-\sin x}{-\cos x}
\]
4. **Simplify the expression:**
\[
\frac{-\sin x}{-\cos x} = \frac{\sin x}{\cos x}
\]
5. **Recognize the resulting trigonometric ratio:**
\[
\frac{\sin x}{\cos x} = \tan x
\]
Thus, we have shown that:
\[
\frac{\cos \left(\frac{\pi}{2} + x\right)}{\cos (\pi + x)} = \tan x
\]
---
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