Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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![### Trigonometric Expression
**Problem 6:**
\[ \cos x (\csc x - \sec x) - \cot x \]
This expression involves trigonometric functions where:
- \(\cos x\) represents the cosine of angle \(x\),
- \(\csc x\) stands for the cosecant of angle \(x\), which is the reciprocal of \(\sin x\),
- \(\sec x\) stands for the secant of angle \(x\), which is the reciprocal of \(\cos x\),
- \(\cot x\) denotes the cotangent of angle \(x\), which is the reciprocal of the tangent of \(x\).
To analyze and possibly simplify this expression further, we can follow a few steps to see if there are trigonometric identities that can simplify the given components.
1. Recall the trigonometric identities:
- \(\csc x = \frac{1}{\sin x}\)
- \(\sec x = \frac{1}{\cos x}\)
- \(\cot x = \frac{\cos x}{\sin x}\)
2. Substitute these identities into the expression:
\[ \cos x \left(\frac{1}{\sin x} - \frac{1}{\cos x}\right) - \frac{\cos x}{\sin x} \]
3. Simplify inside the parenthesis:
\[ \cos x \left(\frac{\cos x - \sin x}{\sin x \cos x}\right) - \frac{\cos x}{\sin x} \]
4. Factor out common terms when possible and continue to simplify.
This problem serves as a practice in manipulating and simplifying trigonometric expressions, reinforcing understanding of fundamental trigonometric identities.](https://content.bartleby.com/qna-images/question/050b7896-96e1-4a53-ae28-da4cd34956eb/238e1094-104e-4608-a2d5-5833c953ecf0/opipcvn_processed.jpeg)
Transcribed Image Text:### Trigonometric Expression
**Problem 6:**
\[ \cos x (\csc x - \sec x) - \cot x \]
This expression involves trigonometric functions where:
- \(\cos x\) represents the cosine of angle \(x\),
- \(\csc x\) stands for the cosecant of angle \(x\), which is the reciprocal of \(\sin x\),
- \(\sec x\) stands for the secant of angle \(x\), which is the reciprocal of \(\cos x\),
- \(\cot x\) denotes the cotangent of angle \(x\), which is the reciprocal of the tangent of \(x\).
To analyze and possibly simplify this expression further, we can follow a few steps to see if there are trigonometric identities that can simplify the given components.
1. Recall the trigonometric identities:
- \(\csc x = \frac{1}{\sin x}\)
- \(\sec x = \frac{1}{\cos x}\)
- \(\cot x = \frac{\cos x}{\sin x}\)
2. Substitute these identities into the expression:
\[ \cos x \left(\frac{1}{\sin x} - \frac{1}{\cos x}\right) - \frac{\cos x}{\sin x} \]
3. Simplify inside the parenthesis:
\[ \cos x \left(\frac{\cos x - \sin x}{\sin x \cos x}\right) - \frac{\cos x}{\sin x} \]
4. Factor out common terms when possible and continue to simplify.
This problem serves as a practice in manipulating and simplifying trigonometric expressions, reinforcing understanding of fundamental trigonometric identities.
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