In Exercises 67–74, rewrite each expression in terms of the given function or functions.cos x---------------- + tan x ; cos x1 + sin x

Verified step by step guidance

1. Identify the expression: \( \frac{\cos x}{1 + \sin x} + \tan x \).
2. Recall that \( \tan x = \frac{\sin x}{\cos x} \). Substitute this into the expression to get \( \frac{\cos x}{1 + \sin x} + \frac{\sin x}{\cos x} \).
3. Find a common denominator for the two fractions, which is \( (1 + \sin x) \cdot \cos x \).
4. Rewrite each fraction with the common denominator: \( \frac{\cos^2 x}{(1 + \sin x) \cdot \cos x} + \frac{\sin x (1 + \sin x)}{(1 + \sin x) \cdot \cos x} \).
5. Combine the fractions: \( \frac{\cos^2 x + \sin x (1 + \sin x)}{(1 + \sin x) \cdot \cos x} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is essential for simplifying trigonometric expressions and rewriting them in terms of other functions.

Reciprocal Functions

Reciprocal functions in trigonometry refer to the relationships between sine, cosine, and their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). For example, sec x is the reciprocal of cos x, and csc x is the reciprocal of sin x. Recognizing these relationships helps in rewriting expressions and solving trigonometric equations.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic techniques to rewrite expressions in a more manageable form. This process often includes factoring, combining fractions, and substituting equivalent trigonometric functions. Mastery of simplification techniques is crucial for solving complex trigonometric problems effectively.