Prove the following identities.
$\(\sec\]\theta\)=\(\frac{1}{\cos\theta}\)$

Verified step by step guidance

1. Start by recalling the definition of the secant function in trigonometry. The secant of an angle \( \theta \) is defined as the reciprocal of the cosine of that angle.
2. Express the secant function in terms of cosine: \( \sec\theta = \frac{1}{\cos\theta} \). This is the fundamental definition of the secant function.
3. To prove the identity \( \sec\theta = \frac{1}{\cos\theta} \), we need to show that this expression holds true for all values of \( \theta \) where \( \cos\theta \neq 0 \).
4. Consider the unit circle, where the cosine of an angle \( \theta \) is the x-coordinate of the point on the circle. The secant, being the reciprocal, represents the ratio of the hypotenuse to the adjacent side in a right triangle.
5. Since \( \sec\theta \) is defined as \( \frac{1}{\cos\theta} \), and this relationship is derived directly from the definitions of the trigonometric functions, the identity \( \sec\theta = \frac{1}{\cos\theta} \) is proven by definition.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and secant, relate the angles of a triangle to the ratios of its sides. The secant function, denoted as sec(θ), is defined as the reciprocal of the cosine function, specifically sec(θ) = 1/cos(θ). Understanding these functions is essential for manipulating and proving identities in trigonometry.

Reciprocal Identities

Reciprocal identities are fundamental relationships in trigonometry that express one trigonometric function in terms of another. For example, the identity sec(θ) = 1/cos(θ) illustrates how the secant function is the reciprocal of the cosine function. Mastery of these identities is crucial for simplifying expressions and proving trigonometric identities.

Proving Trigonometric Identities

Proving trigonometric identities involves demonstrating that two expressions are equivalent using algebraic manipulation and known identities. This process often requires substituting one function for another, applying reciprocal identities, and simplifying the expressions. A solid grasp of trigonometric identities and properties is necessary to effectively prove these relationships.