2. Verify: sin x+tan x %3D sin2x sec x cscx+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°.
Related questions
Question
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![## Trigonometric Identity Verification
**Problem Statement:**
Verify the following trigonometric identity:
\[
\frac{\sin x + \tan x}{\csc x + \cot x} = \sin^2 x \cdot \sec x
\]
**Steps for Verification:**
1. **Rewrite Trigonometric Functions:**
- Recall the definitions of the trigonometric functions:
- \(\tan x = \frac{\sin x}{\cos x}\)
- \(\csc x = \frac{1}{\sin x}\)
- \(\cot x = \frac{\cos x}{\sin x}\)
- \(\sec x = \frac{1}{\cos x}\)
2. **Simplify the Expression:**
- Substitute the above identities into the equation:
- \(\frac{\sin x + \frac{\sin x}{\cos x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}}\)
3. **Common Denominators:**
- Simplify the numerator and the denominator:
- Numerator: \(\sin x + \frac{\sin x}{\cos x} = \frac{\sin x \cos x + \sin x}{\cos x}\)
- Denominator: \(\frac{1 + \cos x}{\sin x}\)
4. **Dividing Fractions:**
- When dividing fractions, multiply by the reciprocal:
- \(\frac{\frac{\sin x (\cos x + 1)}{\cos x}}{\frac{1 + \cos x}{\sin x}} = \frac{\sin x (\cos x + 1)}{\cos x} \cdot \frac{\sin x}{1 + \cos x}\)
5. **Canceling Common Factors:**
- Simplify by canceling \((1 + \cos x)\) from numerator and denominator:
- \(\frac{\sin^2 x}{\cos x} = \sin^2 x \cdot \sec x\)
Thus, the identity has been verified.
**Conclusion:**
The given expression simplifies correctly to the desired result, verifying the identity is true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8ab2c5b-9b7e-4f04-bb69-d806282b0355%2F820614c3-6cef-470f-907a-f8a14f39852c%2Flg31i7o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Trigonometric Identity Verification
**Problem Statement:**
Verify the following trigonometric identity:
\[
\frac{\sin x + \tan x}{\csc x + \cot x} = \sin^2 x \cdot \sec x
\]
**Steps for Verification:**
1. **Rewrite Trigonometric Functions:**
- Recall the definitions of the trigonometric functions:
- \(\tan x = \frac{\sin x}{\cos x}\)
- \(\csc x = \frac{1}{\sin x}\)
- \(\cot x = \frac{\cos x}{\sin x}\)
- \(\sec x = \frac{1}{\cos x}\)
2. **Simplify the Expression:**
- Substitute the above identities into the equation:
- \(\frac{\sin x + \frac{\sin x}{\cos x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}}\)
3. **Common Denominators:**
- Simplify the numerator and the denominator:
- Numerator: \(\sin x + \frac{\sin x}{\cos x} = \frac{\sin x \cos x + \sin x}{\cos x}\)
- Denominator: \(\frac{1 + \cos x}{\sin x}\)
4. **Dividing Fractions:**
- When dividing fractions, multiply by the reciprocal:
- \(\frac{\frac{\sin x (\cos x + 1)}{\cos x}}{\frac{1 + \cos x}{\sin x}} = \frac{\sin x (\cos x + 1)}{\cos x} \cdot \frac{\sin x}{1 + \cos x}\)
5. **Canceling Common Factors:**
- Simplify by canceling \((1 + \cos x)\) from numerator and denominator:
- \(\frac{\sin^2 x}{\cos x} = \sin^2 x \cdot \sec x\)
Thus, the identity has been verified.
**Conclusion:**
The given expression simplifies correctly to the desired result, verifying the identity is true.
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