Verify each identity:

(tan x + tan y) / (1 - tan x tan y) = (sin x cos y + cos x sin y) / (cos x cos y - sin x sin y)

Tan(x) + tan(y)/1 - tan(x)tan(y) = sin(x)cos(y) + cos(x)sin(y)/cos(x)cos(y) - sin(x)sin(y)

tan(x + y) = sin(x + y)/cos(x + y)
tan(x + y) = tan(x + y)

Using double angle identity in trigonometry, sin x cos y + cos x sin y is equal to the sum of x and y that is the angle inside the sine function notation. On the other hand, cos x cos y - sin x sin y is equal to cos (x+y) while (tan x + tan y) / (1 - tan x tan y) is equal to tan (x+y). Since tan (x+y) = sin (x+y)/ cos (x+y), the problem is solved