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. Start by writing down the given expression: \(\cos \beta (\sec \beta + \csc \beta)\).
2. Recall the definitions of the secant and cosecant functions in terms of cosine and sine: \(\sec \beta = \frac{1}{\cos \beta}\) and \(\csc \beta = \frac{1}{\sin \beta}\).
3. Substitute these definitions into the expression to get: \(\cos \beta \left( \frac{1}{\cos \beta} + \frac{1}{\sin \beta} \right)\).
4. Distribute \(\cos \beta\) across the terms inside the parentheses: \(\cos \beta \cdot \frac{1}{\cos \beta} + \cos \beta \cdot \frac{1}{\sin \beta}\).
5. Simplify each term: the first term simplifies to 1, and the second term becomes \(\frac{\cos \beta}{\sin \beta}\). Recognize that \(\frac{\cos \beta}{\sin \beta}\) is \(\cot \beta\). So the simplified expression is \(1 + \cot \beta\).

Key Concepts

Reciprocal Trigonometric Functions

Secant (sec) and cosecant (csc) are reciprocal functions of cosine and sine, respectively. Specifically, sec β = 1/cos β and csc β = 1/sin β. Understanding these relationships helps in rewriting expressions to eliminate quotients.

Distributive Property in Algebra

The distributive property allows multiplication of a single term across terms inside parentheses. For example, cos β(sec β + csc β) expands to cos β·sec β + cos β·csc β, enabling simplification by applying trigonometric identities.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting terms using identities and eliminating fractions or quotients. The goal is to express the result in a simpler or more standard form without division, often by substituting reciprocal identities.