Textbook Question
Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.38
Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
(csc θ sec θ)/cot θ
Verified step by step guidance1
Start by recalling the fundamental trigonometric identities: \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
Substitute these identities into the expression \( \frac{\csc \theta \sec \theta}{\cot \theta} \) to get \( \frac{\frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta}}{\frac{\cos \theta}{\sin \theta}} \).
Simplify the expression by multiplying the numerators and denominators: \( \frac{\frac{1}{\sin \theta \cos \theta}}{\frac{\cos \theta}{\sin \theta}} \).
To simplify further, multiply by the reciprocal of the denominator: \( \frac{1}{\sin \theta \cos \theta} \times \frac{\sin \theta}{\cos \theta} \).
Cancel out the common terms in the numerator and the denominator to simplify the expression to a single trigonometric function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, serve as the foundation for simplifying trigonometric expressions. Understanding these identities is crucial for manipulating and simplifying expressions like (csc θ sec θ)/cot θ.
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Reciprocal Functions
Reciprocal functions in trigonometry refer to pairs of functions that are inverses of each other. For example, cosecant (csc) is the reciprocal of sine (sin), secant (sec) is the reciprocal of cosine (cos), and cotangent (cot) is the reciprocal of tangent (tan). Recognizing these relationships allows for easier simplification of expressions by substituting one function for its reciprocal.
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Quotient Identities
Quotient identities express the relationships between the sine, cosine, and tangent functions. Specifically, tangent is defined as the ratio of sine to cosine (tan θ = sin θ/cos θ), and cotangent is the reciprocal of tangent (cot θ = cos θ/sin θ). Utilizing these identities is essential for simplifying expressions involving cotangent, as seen in the given expression (csc θ sec θ)/cot θ.
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