Perform each indicated operation and simplify the result so that there are no quotients.
cos β(sec β + csc β)

Verified step by step guidance

1. Recognize that \( \sec \beta = \frac{1}{\cos \beta} \) and \( \csc \beta = \frac{1}{\sin \beta} \).
2. Substitute these identities into the expression: \( \cos \beta \left( \frac{1}{\cos \beta} + \frac{1}{\sin \beta} \right) \).
3. Distribute \( \cos \beta \) across the terms inside the parentheses: \( \cos \beta \cdot \frac{1}{\cos \beta} + \cos \beta \cdot \frac{1}{\sin \beta} \).
4. Simplify each term: \( 1 + \frac{\cos \beta}{\sin \beta} \).
5. Recognize that \( \frac{\cos \beta}{\sin \beta} = \cot \beta \), so the expression simplifies to \( 1 + \cot \beta \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as cosine (cos), secant (sec), and cosecant (csc), are fundamental in trigonometry. Cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle. Secant is the reciprocal of cosine, while cosecant is the reciprocal of sine. Understanding these functions is essential for manipulating and simplifying expressions involving angles.

Reciprocal Identities

Reciprocal identities are relationships that express trigonometric functions in terms of their reciprocals. For example, sec β = 1/cos β and csc β = 1/sin β. These identities are crucial for simplifying expressions, as they allow us to rewrite functions in a more manageable form, facilitating operations like addition and multiplication.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves combining and reducing terms to achieve a more straightforward form. This process often includes using identities, factoring, and eliminating quotients. In the given expression, recognizing how to combine sec β and csc β with cos β will lead to a clearer and more concise result, which is a key skill in trigonometry.