Match each expression in Column I with its equivalent expression in Column II.
(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))

Verified step by step guidance

1. Recognize that the given expression \( \frac{\tan(\frac{\pi}{3}) - \tan(\frac{\pi}{4})}{1 + \tan(\frac{\pi}{3}) \tan(\frac{\pi}{4})} \) matches the formula for the tangent of a difference of two angles: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).
2. Identify the angles \( A = \frac{\pi}{3} \) and \( B = \frac{\pi}{4} \) in the expression, so the expression simplifies to \( \tan\left(\frac{\pi}{3} - \frac{\pi}{4}\right) \).
3. Calculate the difference of the angles inside the tangent function: \( \frac{\pi}{3} - \frac{\pi}{4} = \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{\pi}{12} \).
4. Rewrite the original expression as \( \tan\left(\frac{\pi}{12}\right) \), which is the equivalent expression in Column II.
5. If needed, recall or use the exact value or approximation of \( \tan\left(\frac{\pi}{12}\right) \) to verify the equivalence with the expressions in Column II.

Key Concepts

Tangent Function and Its Values

The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. Key values such as tan(π/3) = √3 and tan(π/4) = 1 are often used in trigonometric calculations and simplifications.

Tangent Difference Formula

The tangent difference formula states that (tan A - tan B) / (1 + tan A tan B) equals tan(A - B). This identity is essential for simplifying expressions involving differences of tangents into a single tangent function.

Angle Subtraction in Radians

Understanding how to subtract angles measured in radians, such as π/3 - π/4, is crucial for applying trigonometric identities correctly. This involves finding a common denominator and simplifying the resulting angle before evaluating the tangent.