Determine whether each statement is true or false. If false, tell why. See Example 4.tan² 60° + 1 = sec² 60°

Verified step by step guidance

1. Recall the Pythagorean identity: \( \tan^2 \theta + 1 = \sec^2 \theta \).
2. Substitute \( \theta = 60^\circ \) into the identity: \( \tan^2 60^\circ + 1 = \sec^2 60^\circ \).
3. Calculate \( \tan 60^\circ \) using the known value: \( \tan 60^\circ = \sqrt{3} \).
4. Calculate \( \sec 60^\circ \) using the known value: \( \sec 60^\circ = 2 \).
5. Verify if \( (\sqrt{3})^2 + 1 = 2^2 \) holds true to determine if the statement is true or false.

Key Concepts

Pythagorean Identity

The Pythagorean identity in trigonometry states that for any angle θ, the relationship sin²θ + cos²θ = 1 holds true. This identity is foundational for deriving other trigonometric identities, including the secant and tangent functions. It helps in understanding how the squares of sine and cosine relate to the unit circle.

Tangent and Secant Functions

The tangent function, tan(θ), is defined as the ratio of the opposite side to the adjacent side in a right triangle, or tan(θ) = sin(θ)/cos(θ). The secant function, sec(θ), is the reciprocal of the cosine function, sec(θ) = 1/cos(θ). Understanding these functions is crucial for evaluating trigonometric expressions and verifying identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. One important identity is tan²θ + 1 = sec²θ, which is derived from the Pythagorean identity. Recognizing and applying these identities is essential for simplifying expressions and solving trigonometric equations.