Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Derive the subtraction formula for the tangent function.

Step-by-Step Solution

Answer

\( \tan(A-B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \)

Step 1: Understanding the Goal

We aim to deduce the formula for \( \tan(A-B) \) in terms of \( \tan(A) \) and \( \tan(B) \). This is known as the tangent subtraction formula.

Step 2: Using the Tangent Identity

Recall that \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Hence, \( \tan(A-B) = \frac{\sin(A-B)}{\cos(A-B)} \).

Step 3: Apply Sine and Cosine Difference Identities

Utilize the identities \( \sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B) \) and \( \cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B) \).

Step 4: Substitute the Difference Identities

Insert the sine and cosine difference identities into the tangent definition: \( \tan(A-B) = \frac{\sin(A)\cos(B) - \cos(A)\sin(B)}{\cos(A)\cos(B) + \sin(A)\sin(B)} \).

Step 5: Express in Terms of Tangents

Replace the sine and cosine terms using \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This gives: \( \tan(A) = \frac{\sin(A)}{\cos(A)} \) and \( \tan(B) = \frac{\sin(B)}{\cos(B)} \).

Step 6: Simplification

Substitute \( \sin(A) = \tan(A)\cos(A) \) and \( \sin(B) = \tan(B)\cos(B) \) into the equation: \( \tan(A-B) = \frac{\tan(A)\cos(A)\cos(B) - \tan(B)\cos(A)\cos(B)}{\cos(A)\cos(B) + \tan(A)\tan(B)\cos(A)\cos(B)} \).

Step 7: Final Expression Derivation

Factor out \( \cos(A)\cos(B) \) from both the numerator and the denominator: \( \tan(A-B) = \frac{\cos(A)\cos(B)[\tan(A) - \tan(B)]}{\cos(A)\cos(B)[1 + \tan(A)\tan(B)]} \).

Step 8: Cancellation of Common Terms

Cancel out the common \( \cos(A)\cos(B) \) terms from the numerator and denominator, resulting in: \( \tan(A-B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \). This is the tangent difference formula.

Key Concepts

Tangent Subtraction Formula

The tangent subtraction formula is a fundamental trigonometric identity used to express the tangent of the difference between two angles. In its essence, it allows us to find the value of \( \tan(A-B) \) using the individual tangents \( \tan(A) \) and \( \tan(B) \). This formula is crucial for simplifying complex trigonometric expressions and solving problems that involve angle differences. The subtraction formula for tangent is given by:

• \( \tan(A-B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \)

To derive this, we utilize other trigonometric identities, which involve the sine and cosine of the angle difference. The formula is especially prevalent in calculus and algebraic trigonometry, where it aids in simplifying and solving equations that involve tangent functions.

Sine and Cosine Difference Identities

Sine and cosine difference identities are key tools employed in deriving the tangent subtraction formula. These identities help break down more complex expressions involving angle differences into combinations of sine and cosine functions. The identities include:

• \( \sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B) \)
• \( \cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B) \)

These formulas allow us to substitute expressions for \( \sin(A-B) \) and \( \cos(A-B) \) when developing the tangent subtraction formula. By expressing the tangent as a ratio of sine to cosine, these identities simplify complex trigonometric problems and facilitate deeper understanding of angle relationships.

Trigonometric Simplification

Trigonometric simplification is the process of reducing complex trigonometric expressions to simpler, more manageable forms. This is particularly important when solving equations or proving identities. In the context of the tangent subtraction formula, simplification involves:

• Substituting equivalent expressions using trigonometric identities.
• Factoring common terms in the expression.
• Canceling like terms to derive a cleaner formula.

In our derivation, simplification becomes crucial as we convert the expression \( \frac{\sin(A-B)}{\cos(A-B)} \) into \( \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \). This process highlights the importance of understanding identity transformations and how they lead to more elegant solutions in trigonometric calculations.

Tangent Function

The tangent function is one of the primary trigonometric functions, closely linked with the sine and cosine functions. It is defined as the ratio of the sine to the cosine of an angle: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This unique relationship means that tangent can often be utilized to relate sine and cosine in various formulas, including the subtraction formula.The tangent function is periodic with a period of \( \pi \), meaning it repeats its values over intervals of \( \pi \). It is also important to note that tangent has asymptotes where \( \cos(x) = 0 \) because it makes the function undefined at those points. When applying the subtraction formula, the properties of the tangent function ensure that we can handle angle differences effectively by leveraging its relationship with sine and cosine.