Prove the identity. sin(2x) 2 tan(x) 1 + tan²(x) Write the more complicated side of the equation in terms of sin(x) and cos(x). Simplify the complex fraction. sin(x) 2 tan(x) 1 + tan²(x) = 2. 1 + 2 1+ 2. sin(x) sin(x) cos(x) sin²(x) cos²(x) cos(x) sin (2x) cos²(x) cos²(x) + sin²(x) 2 sin(x) cos(x) cos²(x) sin²(x) X
Prove the identity. sin(2x) 2 tan(x) 1 + tan²(x) Write the more complicated side of the equation in terms of sin(x) and cos(x). Simplify the complex fraction. sin(x) 2 tan(x) 1 + tan²(x) = 2. 1 + 2 1+ 2. sin(x) sin(x) cos(x) sin²(x) cos²(x) cos(x) sin (2x) cos²(x) cos²(x) + sin²(x) 2 sin(x) cos(x) cos²(x) sin²(x) 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°.
Related questions
Question
![Prove the identity.
\[
\frac{\sin(2x)}{1 + \tan^2(x)} = \frac{2 \tan(x)}{1 + \tan^2(x)}
\]
Write the more complicated side of the equation in terms of \(\sin(x)\) and \(\cos(x)\). Simplify the complex fraction.
\[
\frac{2 \tan(x)}{1 + \tan^2(x)} = \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin(x)}{\cos^2(x)}}
\]
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin^2(x)}{\cos^2(x)}}
\]
Marked with a green check (\(\sin(x)/\cos(x)\)) indicates correct steps, while red crosses indicate incorrect placements or notations.
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{\cos^2(x)/\sin^2(x)}
\]
\[
= \frac{2 \cdot \sin(x) \cdot \cos(x)}{\cos^2(x) + \sin^2(x)}
\]
\[
= \frac{2 \sin(x) \cos(x)}{1}
\]
\[
= \sin(2x)
\]
In this solution, the identity for \(\sin(2x) = 2 \sin(x) \cos(x)\) is used to simplify and confirm the trigonometric identity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e308213-df48-4360-8e9d-bfd0a573e9aa%2F0f2a8709-c340-45ef-9b52-9d271d4dbd0b%2Fqorwut4_processed.png&w=3840&q=75)
Transcribed Image Text:Prove the identity.
\[
\frac{\sin(2x)}{1 + \tan^2(x)} = \frac{2 \tan(x)}{1 + \tan^2(x)}
\]
Write the more complicated side of the equation in terms of \(\sin(x)\) and \(\cos(x)\). Simplify the complex fraction.
\[
\frac{2 \tan(x)}{1 + \tan^2(x)} = \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin(x)}{\cos^2(x)}}
\]
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin^2(x)}{\cos^2(x)}}
\]
Marked with a green check (\(\sin(x)/\cos(x)\)) indicates correct steps, while red crosses indicate incorrect placements or notations.
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{\cos^2(x)/\sin^2(x)}
\]
\[
= \frac{2 \cdot \sin(x) \cdot \cos(x)}{\cos^2(x) + \sin^2(x)}
\]
\[
= \frac{2 \sin(x) \cos(x)}{1}
\]
\[
= \sin(2x)
\]
In this solution, the identity for \(\sin(2x) = 2 \sin(x) \cos(x)\) is used to simplify and confirm the trigonometric identity.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning