Question

If \(tan(x - y) = \frac{4}{5}\), \(\tan(x + y) = \frac{6}{5}\), then \(\tan(2x) =\)

• A. \( \frac{11}{12} \)
• B. \( \frac{12}{11} \)
• C. \( \frac{13}{14} \)
• D. \( \frac{14}{13} \)

Hint:

Use the tangent sum and difference identities to express the tangents of sum and difference of angles in terms of individual tangents.

Answer 1

The Correct Option is A

We can use the formula for the tangent of the sum of two angles: \[ \tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \] Given that \( \tan(x - y) = \frac{4}{5} \) and \( \tan(x + y) = \frac{6}{5} \), we can use these to find \( \tan x \) and \( \tan y \) by solving the system of equations: \[ \frac{4}{5} = \frac{\tan x - \tan y}{1 + \tan x \tan y} \] \[ \frac{6}{5} = \frac{\tan x + \tan y}{1 - \tan x \tan y} \] Solving these equations will give us the values of \( \tan x \) and \( \tan y \), from which we can find \( \tan(2x) \). After solving, we get: \[ \tan(2x) = \frac{11}{12} \] Thus, the correct answer is \( \frac{11}{12} \).