Verify that each equation is an identity. See Example 4.
tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)

검증된 단계별 안내

Recall the tangent subtraction formula: \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\). This will be useful to expand both \(\tan(x - y)\) and \(\tan(y - x)\).

Apply the formula to \(\tan(x - y)\): write it as \(\frac{\tan x - \tan y}{1 + \tan x \tan y}\).

Apply the formula to \(\tan(y - x)\): write it as \(\frac{\tan y - \tan x}{1 + \tan y \tan x}\). Note that \(1 + \tan y \tan x\) is the same as \(1 + \tan x \tan y\) due to commutativity.

Substitute these expressions back into the left side of the equation: \(\tan(x - y) - \tan(y - x) = \frac{\tan x - \tan y}{1 + \tan x \tan y} - \frac{\tan y - \tan x}{1 + \tan x \tan y}\).

Since the denominators are the same, combine the fractions and simplify the numerator: \(\frac{(\tan x - \tan y) - (\tan y - \tan x)}{1 + \tan x \tan y}\). Simplify the numerator carefully to show it equals \(2(\tan x - \tan y)\), which matches the right side of the identity.

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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 using known formulas like angle sum and difference identities.

Tangent Difference Formula

The tangent difference formula states that tan(a - b) = (tan a - tan b) / (1 + tan a tan b). This formula is essential for rewriting and simplifying expressions involving tangent of differences, which is key to verifying the given identity.

Algebraic Manipulation of Fractions

Simplifying trigonometric expressions often requires careful algebraic manipulation, including combining fractions, factoring, and simplifying complex rational expressions. Mastery of these skills helps in transforming one side of the identity to match the other.

Verify that each equation is an identity. See Example 4.tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)