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

검증된 단계별 안내

Identify the given expression: \( (\sec x + \csc x)(\sin x + \cos x) - 2 - \cot x \) and the function to express it in terms of: \( \tan x \).

Recall the definitions of the trigonometric functions involved: \( \sec x = \frac{1}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), and \( \cot x = \frac{\cos x}{\sin x} \).

Expand the product \( (\sec x + \csc x)(\sin x + \cos x) \) by distributing each term: \( \sec x \cdot \sin x + \sec x \cdot \cos x + \csc x \cdot \sin x + \csc x \cdot \cos x \).

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 \( \csc x \cdot \sin x = 1 \).

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 \( \tan x \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.