In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x

Verified step by step guidance

1. Identify the given expression: \( (\sec x + \csc x)(\sin x + \cos x) - 2 - \cot x \) and the function to express it in terms of: \( \tan x \).
2. Recall the definitions of the trigonometric functions involved: \( \sec x = \frac{1}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), and \( \cot x = \frac{\cos x}{\sin x} \).
3. Expand the product \( (\sec x + \csc x)(\sin x + \cos x) \) by distributing each term: \( \sec x \cdot \sin x + \sec x \cdot \cos x + \csc x \cdot \sin x + \csc x \cdot \cos x \).
4. Substitute the definitions into each term and simplify where possible, for example, \( \sec x \cdot \sin x = \frac{\sin x}{\cos x} = \tan x \), and \( \csc x \cdot \sin x = 1 \).
5. Combine all simplified terms, subtract 2 and \( \cot x \), then rewrite any remaining expressions in terms of \( \tan x \) using identities such as \( \cot x = \frac{1}{\tan x} \) to express the entire expression solely in terms of \( \tan x \).

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal functions are the inverses of the basic sine, cosine, and tangent functions. Secant (sec x) is 1/cos x, cosecant (csc x) is 1/sin x, and cotangent (cot x) is 1/tan x. Understanding these allows rewriting expressions involving sec, csc, and cot in terms of sine, cosine, or tangent.

Algebraic Manipulation of Trigonometric Expressions

Simplifying trigonometric expressions often requires factoring, expanding, and combining like terms. Recognizing common denominators and using identities helps rewrite complex expressions in terms of specified functions, such as expressing everything in terms of tan x or sin x and cos x.

Pythagorean and Quotient Identities

Pythagorean identities like sin²x + cos²x = 1 and quotient identities such as tan x = sin x / cos x are essential tools. They enable conversion between different trigonometric functions and simplify expressions by substituting equivalent forms.