We can use the formula for tangent of a sum of angles to solve this problem:
tan(x + y) = (tan x + tan y) / (1 - tan x * tan y)
Substituting the given values, we get:
4/3 = (5/13 + tan y) / (1 - 5/13 * tan y)
Multiplying both sides by the denominator and simplifying, we get:
52/3 - 20/3 * tan y = 5/13 + tan y
Multiplying both sides by 3/20 and simplifying, we get:
tan y = (52/3 - 5/13) / (20/3 + 1)
tan y = (1699/39) / (23/3)
tan y = 221/299
Therefore, tan y is approximately 0.7391 (rounded to four decimal places).