In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator.sec² 23° - tan² 23°

Verified step by step guidance

1. Recognize that the expression \( \sec^2 \theta - \tan^2 \theta \) is a trigonometric identity.
2. Recall the Pythagorean identity: \( \sec^2 \theta = 1 + \tan^2 \theta \).
3. Substitute \( \sec^2 \theta = 1 + \tan^2 \theta \) into the expression \( \sec^2 23^\circ - \tan^2 23^\circ \).
4. Simplify the expression: \( (1 + \tan^2 23^\circ) - \tan^2 23^\circ \).
5. Conclude that the expression simplifies to 1.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the relationship sin²θ + cos²θ = 1 holds true. This identity is fundamental in trigonometry as it connects the sine and cosine functions, allowing for the derivation of other identities and simplifications of expressions involving trigonometric functions.

Secant and Tangent Functions

The secant function, sec(θ), is defined as the reciprocal of the cosine function, while the tangent function, tan(θ), is the ratio of the sine to the cosine function. Understanding these functions is crucial for manipulating expressions involving them, especially when applying identities to simplify or evaluate trigonometric expressions.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. The identity sec²θ - tan²θ = 1 is a specific example that arises from the Pythagorean identity, and it is essential for solving problems that require the evaluation of expressions without a calculator.