Perform each indicated operation and simplify the result so that there are no quotients.
cos x/sec x + sin x/csc x

Verified step by step guidance

1. Recall the definitions of the reciprocal trigonometric functions: \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\).
2. Rewrite each quotient using these definitions: \(\frac{\cos x}{\sec x} = \cos x \times \cos x\) and \(\frac{\sin x}{\csc x} = \sin x \times \sin x\).
3. Simplify the expressions by multiplying: \(\cos x \times \cos x = \cos^{2} x\) and \(\sin x \times \sin x = \sin^{2} x\).
4. Add the two simplified terms together: \(\cos^{2} x + \sin^{2} x\).
5. Use the Pythagorean identity \(\sin^{2} x + \cos^{2} x = 1\) to simplify the expression to its simplest form.

Key Concepts

Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as sec x = 1/cos x and csc x = 1/sin x. Understanding these allows you to rewrite expressions like cos x/sec x and sin x/csc x without fractions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves combining terms, eliminating complex fractions, and rewriting functions in simpler forms. This process often uses algebraic manipulation and identities to express the result without quotients.

Basic Trigonometric Functions

Familiarity with sine, cosine, secant, and cosecant functions and their properties is essential. Knowing their definitions and relationships helps in transforming and simplifying expressions involving these functions.