7. Prove the following trigonometric identity: secx-2sin x = (sin x − cosx)2 cosx
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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![### Prove the Following Trigonometric Identity
**Problem Statement:**
Prove the following trigonometric identity:
\[ \sec x - 2 \sin x = \frac{(\sin x - \cos x)^2}{\cos x} \]
**Detailed Explanation:**
To prove this identity, let's start by expressing \(\sec x\) in terms of \(\cos x\).
1. Remember that \(\sec x = \frac{1}{\cos x}\).
2. Substitute \(\sec x\) in the given identity:
\[ \frac{1}{\cos x} - 2 \sin x = \frac{(\sin x - \cos x)^2}{\cos x} \]
**Next Steps:**
1. Combine the terms on the left-hand side over a common denominator:
\[ \frac{1 - 2 \sin x \cos x}{\cos x} \]
2. Expand the numerator on the right-hand side:
\[ \frac{(\sin x - \cos x)^2}{\cos x} = \frac{\sin^2 x - 2 \sin x \cos x + \cos^2 x}{\cos x} \]
3. Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):
\[ \frac{1 - 2 \sin x \cos x}{\cos x} = \frac{1 - 2 \sin x \cos x + \cos^2 x + \sin^2 x - \cos^2 x}{\cos x} \]
Since \(1 - 2 \sin x \cos x\) equals both sides, the identity is verified.
This confirms the left-hand side equals the right-hand side, thus proving the given trigonometric identity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd88595c7-c499-4476-ae94-4573188e2c46%2F07293b61-de9b-4941-84c6-ec4e734b7ade%2Fy80ku6n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Prove the Following Trigonometric Identity
**Problem Statement:**
Prove the following trigonometric identity:
\[ \sec x - 2 \sin x = \frac{(\sin x - \cos x)^2}{\cos x} \]
**Detailed Explanation:**
To prove this identity, let's start by expressing \(\sec x\) in terms of \(\cos x\).
1. Remember that \(\sec x = \frac{1}{\cos x}\).
2. Substitute \(\sec x\) in the given identity:
\[ \frac{1}{\cos x} - 2 \sin x = \frac{(\sin x - \cos x)^2}{\cos x} \]
**Next Steps:**
1. Combine the terms on the left-hand side over a common denominator:
\[ \frac{1 - 2 \sin x \cos x}{\cos x} \]
2. Expand the numerator on the right-hand side:
\[ \frac{(\sin x - \cos x)^2}{\cos x} = \frac{\sin^2 x - 2 \sin x \cos x + \cos^2 x}{\cos x} \]
3. Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):
\[ \frac{1 - 2 \sin x \cos x}{\cos x} = \frac{1 - 2 \sin x \cos x + \cos^2 x + \sin^2 x - \cos^2 x}{\cos x} \]
Since \(1 - 2 \sin x \cos x\) equals both sides, the identity is verified.
This confirms the left-hand side equals the right-hand side, thus proving the given trigonometric identity.
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