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

The equation you've provided is the formula for the tangent of the sum of two angles, \( A \) and \( B \). This formula is useful in trigonometry for simplifying expressions involving the tangent function. To understand this formula better, let's break it down step by step: 1. **Understanding the Components**: - \( \tan A \) and \( \tan B \) are the tangents of angles \( A \) and \( B \), respectively. - The right side of the equation combines these two tangents in a specific way. 2. **Formula Explanation**: - The left side, \( \tan (A + B) \), represents the tangent of the sum of angles \( A \) and \( B \). - The right side, \( \frac{\tan A + \tan B}{1 - \tan A \tan B} \), shows how to compute this tangent using the individual tangents of \( A \) and \( B \). 3. **Conditions for Validity**: - This formula is valid as long as \( 1 - \tan A \tan B \neq 0 \). If \( \tan A \tan B = 1 \), the formula would involve division by zero, which is undefined. 4. **Applications**: - This formula is often used in problems involving angle addition in trigonometry, such as solving equations or simplifying expressions. If you have a specific problem or application in mind regarding this formula, please provide more details, and I can assist you further!