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 involving tangent and secant: \(\tan^2 \theta + 1 = \sec^2 \theta\). This identity holds true for any angle \(\theta\) where these functions are defined.
2. Substitute \(\theta = 60^\circ\) into the identity to check if the statement holds: \(\tan^2 60^\circ + 1 = \sec^2 60^\circ\).
3. Calculate \(\tan 60^\circ\) using the known exact value: \(\tan 60^\circ = \sqrt{3}\), so \(\tan^2 60^\circ = (\sqrt{3})^2 = 3\).
4. Calculate \(\sec 60^\circ\) using the definition \(\sec \theta = \frac{1}{\cos \theta}\) and the known value \(\cos 60^\circ = \frac{1}{2}\), so \(\sec 60^\circ = 2\) and \(\sec^2 60^\circ = 2^2 = 4\).
5. Compare the two sides: \(\tan^2 60^\circ + 1 = 3 + 1 = 4\) and \(\sec^2 60^\circ = 4\). Since both sides are equal, the statement is true.

Key Concepts

Pythagorean Identity involving Tangent and Secant

The identity tan²θ + 1 = sec²θ is a fundamental trigonometric identity derived from the Pythagorean theorem. It relates the square of the tangent function and the secant function for any angle θ, and is essential for verifying equations involving these functions.

Evaluating Trigonometric Functions at Specific Angles

To verify the given statement, one must accurately calculate tan 60° and sec 60°. Knowing exact values of trigonometric functions at common angles like 30°, 45°, and 60° is crucial for precise evaluation and comparison.

Definition of Secant Function

The secant function, sec θ, is defined as the reciprocal of the cosine function: sec θ = 1/cos θ. Understanding this definition helps in computing sec 60° and comparing it with expressions involving tangent.