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6. cos x(csc x secx) secx) cotx

6. cos x(csc x secx) secx) cotx

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 5GP
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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.
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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