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.
(sec²θ - 1)/(csc²θ - 1)

Verified step by step guidance

1. Recognize that \( \sec^2\theta - 1 \) can be rewritten using the Pythagorean identity: \( \sec^2\theta = 1 + \tan^2\theta \), so \( \sec^2\theta - 1 = \tan^2\theta \).
2. Similarly, use the Pythagorean identity for cosecant: \( \csc^2\theta = 1 + \cot^2\theta \), so \( \csc^2\theta - 1 = \cot^2\theta \).
3. Rewrite \( \tan^2\theta \) and \( \cot^2\theta \) in terms of sine and cosine: \( \tan^2\theta = \left(\frac{\sin\theta}{\cos\theta}\right)^2 \) and \( \cot^2\theta = \left(\frac{\cos\theta}{\sin\theta}\right)^2 \).
4. Substitute these expressions back into the original expression: \( \frac{\tan^2\theta}{\cot^2\theta} = \frac{\left(\frac{\sin\theta}{\cos\theta}\right)^2}{\left(\frac{\cos\theta}{\sin\theta}\right)^2} \).
5. Simplify the expression by multiplying the numerator and the denominator: \( \left(\frac{\sin\theta}{\cos\theta}\right)^2 \times \left(\frac{\sin\theta}{\cos\theta}\right)^2 = \left(\frac{\sin^2\theta}{\cos^2\theta}\right) \times \left(\frac{\sin^2\theta}{\cos^2\theta}\right) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is essential for rewriting trigonometric expressions in terms of sine and cosine, as they provide the foundational relationships between different functions.

Reciprocal Functions

Reciprocal functions in trigonometry refer to pairs of functions that are inverses of each other, such as sine and cosecant, or cosine and secant. For example, sec(θ) = 1/cos(θ) and csc(θ) = 1/sin(θ). Recognizing these relationships allows for the conversion of sec²θ and csc²θ into expressions involving sine and cosine, which is crucial for simplifying the given expression.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves rewriting complex expressions into simpler forms, often by eliminating quotients and combining like terms. This process typically uses identities and algebraic manipulation. In the context of the given expression, it requires transforming sec²θ and csc²θ into sine and cosine terms, allowing for a clearer and more manageable expression.