In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{\cos x}{1+\sin x}+\tan x$ ; $\cos x$

Verified step by step guidance

1. Start by rewriting the given expression clearly: \(\frac{\cos x}{1 + \sin x} + \tan x \cdot \cos x\).
2. Recall that \(\tan x = \frac{\sin x}{\cos x}\), so substitute \(\tan x\) in the expression to get \(\frac{\cos x}{1 + \sin x} + \frac{\sin x}{\cos x} \cdot \cos x\).
3. Simplify the second term by canceling \(\cos x\) in numerator and denominator: \(\frac{\cos x}{1 + \sin x} + \sin x\).
4. To combine the terms, express \(\sin x\) with a common denominator \(1 + \sin x\): write \(\sin x = \frac{\sin x (1 + \sin x)}{1 + \sin x}\).
5. Now add the two fractions: \(\frac{\cos x}{1 + \sin x} + \frac{\sin x (1 + \sin x)}{1 + \sin x} = \frac{\cos x + \sin x (1 + \sin x)}{1 + \sin x}\). This is the expression rewritten in terms of sine and cosine functions.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow rewriting expressions in different forms, such as converting between sine, cosine, and tangent. Common identities include Pythagorean identities and quotient identities, which are essential for simplifying or rewriting expressions.

Simplifying Complex Fractions

Simplifying complex fractions involves rewriting expressions with fractions in the numerator and denominator into simpler forms. This often requires finding common denominators, factoring, or multiplying numerator and denominator by conjugates. Mastery of this skill helps in expressing trigonometric expressions in terms of a single function or simpler combinations.

Quotient and Reciprocal Relationships

The quotient identities relate tangent and cotangent to sine and cosine, such as tan x = sin x / cos x. Reciprocal identities express functions like sec x and csc x as reciprocals of cosine and sine, respectively. Understanding these relationships is crucial for rewriting expressions involving multiple trigonometric functions into a desired form.