Help! Verify the Identity (Trigonometry functions)

(sinxcosy + cosxsiny) / (cosxcosy - sinxsiny) = (tanx + tany) / (1-tanxtany)

Help me with this problem please!

If you haven't memorized dozens of rules then you can use the following approach, although it's longer:

Working on the right-hand side:

(tanx + tany) / (1-tanxtany) =

[sin(x)/cos(x) + sin(y)/cos(y)] / [1 - (sin(x)/cos(x))*(sin(y)/cos(y))] =

{[sin(x)cos(y) + sin(y)cos(x)] / [cos(x)cos(y)]} / {1 - [sin(x)sin(y)] / [cos(x)cos(y)]} =

{[sin(x)cos(y) + sin(y)cos(x)] / [cos(x)cos(y)]} / {[cos(x)cos(y) - sin(x)sin(y)] / [cos(x)cos(y)]} =

That's a division of two fractions; each fraction is being divided by cos(x)cos(y) so that gets canceled out:

[sin(x)cos(y) + sin(y)cos(x)] / [cos(x)cos(y) - sin(x)sin(y)]

So we have arrived to the left-hand side, which proves the identity. Not as fancy but only primitive algebra is required really, and almost no trigonometry theory is necessary. Good luck...

sin(a)cos(b) + sin(b)cos(a) => sin(a + b)
cos(a)cos(b) - sin(a)sin(b) => cos(a + b)

So we have:

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

tan(x + y) =>
(tan(x) + tan(y)) / (1 - tan(x)tan(y))