Help with PreCalc: Sum/Difference Identities

[Thinkaholic]

Hi, I know this is baby/fetus/sperm/molecule math for all of you, but I'm 13 and trying to self study my way to physics. Anyway, I'm teaching myself pre calculus. Most of it is pretty easy, but I've been stuck on sum/difference identities. It seems clear, and I'm folliwing everything the textbook is telling me. However, I think I'm making stupid mistakes. For example:

Question: tan(4pi/3+5pi/4)

I worked out the format of the tangent sum of two angles, and:
(Tan(4pi/3)+tan(5pi/4) )/(1-tan(4pi/3)tan(5pi/4))
=(sqrt(3)+1)/(1-(sqrt(3)))
That was my answer

Book answer:-sqrt(3)-2

Please tell me what stupid mistake I'm making if you can. I'm probably really embarrassing myself.

 

[mfb]

Try expanding your fraction with (1+sqrt(3)), then you can simplify it further.
In general, you can get rid of nearly all square roots in denominators with this trick.

 

[Mark44]

To elaborate on what @mfb said, multiply your answer by 1, in the form of $\frac{1+\sqrt 3}{1 + \sqrt 3}$. After you simplify, you will get the book's answer.
The trick involves multiplying the denominator by its conjugate -- the same two terms but with the opposite sign between them. If you multiply the numerator by the same quantity, you are multiplying by 1, which doesn't change the actual value of the fraction, but puts it into a different form.

 

[Thinkaholic]

Thanks people! I solved it yesterday, but forgot to say so. :p