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

Verified step by step guidance

1. Start by rewriting the given expression clearly: \( \frac{\tan x + \cot x}{\cos x \cdot \csc x} \).
2. Recall the definitions of the trigonometric functions involved: \( \tan x = \frac{\sin x}{\cos x} \), \( \cot x = \frac{\cos x}{\sin x} \), and \( \csc x = \frac{1}{\sin x} \).
3. Rewrite the numerator \( \tan x + \cot x \) as \( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \). Find a common denominator to combine these two terms.
4. Rewrite the denominator \( \cos x \cdot \csc x \) as \( \cos x \cdot \frac{1}{\sin x} = \frac{\cos x}{\sin x} \).
5. Now, express the entire fraction as \( \frac{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}}{\frac{\cos x}{\sin x}} \). Simplify this complex fraction by multiplying numerator and denominator appropriately to eliminate the complex fraction.

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal functions relate pairs like sine and cosecant, cosine and secant, tangent and cotangent. For example, csc x = 1/sin x and cot x = 1/tan x. Understanding these relationships helps rewrite expressions by substituting one function with its reciprocal.

Simplifying Complex Fractions

Complex fractions involve a fraction divided by another fraction or expression. Simplifying requires rewriting the numerator and denominator in terms of common functions, then multiplying by the reciprocal of the denominator to simplify the overall expression.

Expressing Trigonometric Expressions in Terms of a Given Function

This involves rewriting all parts of an expression using only the specified trigonometric function(s). For example, expressing tan x and cot x in terms of sin x and cos x, or rewriting everything in terms of cos x and csc x, to meet the problem's requirements.