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

Verified step by step guidance

1. Recall the tangent addition formula: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\).
2. Substitute \(\tan(\alpha + \beta)\) in the left-hand side (LHS) of the given equation with the formula: \(\frac{\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} - \tan \beta}{1 + \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \cdot \tan \beta}\).
3. Simplify the numerator by combining the terms over a common denominator: \(\frac{\tan \alpha + \tan \beta - \tan \beta (1 - \tan \alpha \tan \beta)}{1 - \tan \alpha \tan \beta}\).
4. 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 \(1 - \tan \alpha \tan \beta\).
5. 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.

Key Concepts

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.