Verify that each equation is an identity.
(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

검증된 단계별 안내

Recall the tangent addition formula: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\).

Substitute \(\tan(\alpha + \beta)\) in the left-hand side (LHS) of the given equation with the formula: \(\frac{\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} - \tan \beta}{1 + \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \cdot \tan \beta}\).

Simplify the numerator by combining the terms over a common denominator: \(\frac{\tan \alpha + \tan \beta - \tan \beta (1 - \tan \alpha \tan \beta)}{1 - \tan \alpha \tan \beta}\).

Simplify the denominator similarly by combining terms: \(1 + \frac{(\tan \alpha + \tan \beta) \tan \beta}{1 - \tan \alpha \tan \beta}\), then write it as a single fraction over the common denominator \(1 - \tan \alpha \tan \beta\).

After simplifying numerator and denominator, divide the numerator by the denominator (which is equivalent to multiplying numerator by the reciprocal of denominator), and simplify the expression to show it equals \(\tan \alpha\), thus verifying the identity.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often by applying known formulas or algebraic manipulation.

Tangent Addition Formula

The tangent addition formula states that tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula is essential for expressing the tangent of a sum in terms of individual tangents, enabling simplification and verification of identities involving sums of angles.

Algebraic Manipulation of Fractions

Simplifying complex fractions and expressions requires careful algebraic manipulation, such as combining terms, factoring, and canceling common factors. Mastery of these skills is crucial to transform one side of the identity into the other and verify the equation.

Verify that each equation is an identity.(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α