Evaluate each expression. Give exact values. sec² 300° - 2 cos² 150°

Verified step by step guidance

1. Recall the definitions and identities involved: \( \sec \theta = \frac{1}{\cos \theta} \) and therefore \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \). Also, remember the Pythagorean identity \( \sec^2 \theta = 1 + \tan^2 \theta \), but here it is easier to work directly with cosine values.
2. Evaluate \( \cos 300^\circ \) and \( \cos 150^\circ \) using the unit circle or reference angles. For \( 300^\circ \), find the cosine value in the fourth quadrant, and for \( 150^\circ \), find the cosine value in the second quadrant.
3. Square the cosine values to get \( \cos^2 300^\circ \) and \( \cos^2 150^\circ \). Then compute \( \sec^2 300^\circ = \frac{1}{\cos^2 300^\circ} \).
4. Substitute these values into the expression \( \sec^2 300^\circ - 2 \cos^2 150^\circ \) and simplify the expression step-by-step, keeping the exact values (like fractions or square roots) without converting to decimals.
5. Combine like terms and simplify the expression fully to get the exact value of the original expression.

Key Concepts

Trigonometric Functions and Their Values

Understanding the basic trigonometric functions such as cosine and secant is essential. Secant is the reciprocal of cosine, so sec²θ = 1/cos²θ. Knowing how to find exact values of these functions at specific angles, especially those related to special angles and their positions on the unit circle, is crucial.

Unit Circle and Angle Reference

The unit circle helps determine the exact values of trigonometric functions for angles like 150° and 300°. Recognizing the quadrant in which the angle lies allows you to assign the correct sign to the function values. For example, 150° is in the second quadrant where cosine is negative, and 300° is in the fourth quadrant where cosine is positive.

Pythagorean Identities and Simplification

Using identities such as sec²θ = 1 + tan²θ or the Pythagorean identity sin²θ + cos²θ = 1 can simplify expressions. In this problem, rewriting sec² 300° in terms of cosine and combining like terms helps in evaluating the expression exactly. Simplification skills are key to obtaining the final exact value.