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

Verified step by step guidance

1. Start by expressing each trigonometric function in terms of sine and cosine: \( \csc \theta = \frac{1}{\sin \theta} \), \( \cos \theta = \cos \theta \), and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
2. Substitute these expressions into the original expression: \( \csc \theta \cos \theta \tan \theta = \left( \frac{1}{\sin \theta} \right) \cos \theta \left( \frac{\sin \theta}{\cos \theta} \right) \).
3. Simplify the expression by canceling out \( \sin \theta \) and \( \cos \theta \) where possible: \( \frac{1}{\sin \theta} \cdot \cos \theta \cdot \frac{\sin \theta}{\cos \theta} = 1 \).
4. Notice that the \( \sin \theta \) in the numerator and denominator cancel each other out, as do the \( \cos \theta \) terms.
5. The simplified expression is \( 1 \), which is a constant and does not depend on \( \theta \).

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(θ), is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). Understanding this function is crucial for rewriting expressions involving csc in terms of sine and cosine, as it allows us to express all trigonometric functions in a consistent format.

Tangent Function

The tangent function, represented as tan(θ), is defined as the ratio of the sine and cosine functions: tan(θ) = sin(θ)/cos(θ). This relationship is essential for simplifying expressions that include tan, as it enables the conversion of tangent into sine and cosine, facilitating the elimination of quotients in the final expression.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities, such as sin²(θ) + cos²(θ) = 1, help in simplifying expressions by allowing substitutions that eliminate quotients and express all functions in terms of sine and cosine, which is the goal of the problem.