Derive a formula for tan (A − B).

Verified step by step guidance

1. Start by recalling the identity for tangent of a difference: \( \tan(A - B) = \frac{\sin(A - B)}{\cos(A - B)} \).
2. Use the angle subtraction identities for sine and cosine: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \) and \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).
3. Substitute these identities into the formula for \( \tan(A - B) \): \( \tan(A - B) = \frac{\sin A \cos B - \cos A \sin B}{\cos A \cos B + \sin A \sin B} \).
4. To simplify, divide both the numerator and the denominator by \( \cos A \cos B \): \( \tan(A - B) = \frac{\frac{\sin A}{\cos A} - \frac{\sin B}{\cos B}}{1 + \frac{\sin A}{\cos A} \cdot \frac{\sin B}{\cos B}} \).
5. Recognize that \( \frac{\sin A}{\cos A} = \tan A \) and \( \frac{\sin B}{\cos B} = \tan B \), leading to the final formula: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving equations in trigonometry. Key identities include the Pythagorean identities, reciprocal identities, and angle sum/difference identities, which are crucial for deriving formulas like tan(A - B).

Angle Difference Formula

The angle difference formula for tangent states that tan(A - B) can be expressed in terms of the tangents of angles A and B. Specifically, it is given by the formula tan(A - B) = (tan A - tan B) / (1 + tan A * tan B). This formula is derived from the sine and cosine definitions of tangent and is fundamental for solving problems involving the difference of angles.

Derivation Techniques

Derivation techniques in calculus involve using algebraic manipulation and known identities to derive new formulas or results. In the context of trigonometric functions, this often includes substituting known identities and simplifying expressions. Mastery of these techniques is essential for effectively deriving formulas like tan(A - B) and understanding their applications in various mathematical contexts.