Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
csc θ cos θ tan θ

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Recall the definitions of the trigonometric functions in terms of sine and cosine: \( \csc \theta = \frac{1}{\sin \theta} \), \( \cos \theta = \cos \theta \), and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Rewrite the expression \( \csc \theta \cos \theta \tan \theta \) by substituting these definitions: \( \left( \frac{1}{\sin \theta} \right) \cdot \cos \theta \cdot \left( \frac{\sin \theta}{\cos \theta} \right) \).

Combine the terms by multiplying the fractions and functions: \( \frac{1}{\sin \theta} \times \cos \theta \times \frac{\sin \theta}{\cos \theta} = \frac{1}{\sin \theta} \times \frac{\cos \theta \sin \theta}{\cos \theta} \).

Simplify the expression by canceling common factors in numerator and denominator: \( \frac{1}{\sin \theta} \times \frac{\cos \theta \sin \theta}{\cos \theta} = \frac{1}{\sin \theta} \times \sin \theta = 1 \).

Confirm that the simplified expression contains no quotients and only functions of \( \theta \), completing the simplification.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Reciprocal and Quotient Identities

Trigonometric functions like cosecant (csc) and tangent (tan) can be expressed in terms of sine and cosine using reciprocal and quotient identities. Specifically, csc θ = 1/sin θ and tan θ = sin θ/cos θ. These identities allow rewriting expressions to involve only sine and cosine.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves combining terms, canceling common factors, and eliminating quotients to write the expression in a more compact form. This process often requires substituting identities and carefully manipulating fractions to achieve a form with only sine and cosine functions and no division.

Function Domain and Variable Consistency

When rewriting expressions, it is important to ensure all functions depend on the same variable (θ in this case) and that the domain restrictions are considered. This consistency avoids confusion and ensures the simplified expression accurately represents the original function for all valid θ values.

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sec² θ - 1

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Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sin θ - cos θ) (csc θ + sec θ)

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Factor each trigonometric expression.

sin³ α + cos³ α

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Simplify each expression.

±√[(1 - cos 8θ)/(1 + cos 8θ)]

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Perform each indicated operation and simplify the result so that there are no quotients.

(1 + tan θ)² - 2 tan θ

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