How do I prove the addition formula for $\tan(a+b)$?
I have $$\displaystyle \frac{\sin(a+b)}{\cos(a+b)}=\frac{\sin(a)\cos(b)+\cos(a)\sin(b)}{\cos(a)\cos(b)-\sin(a)\sin(b)}$$ but that's way more complicated than the result I need, which is $$\displaystyle \frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}$$
Am I on the right track? What should I do next?
You are almost there. Notice that you have two terms on the top and bottom, and so does the answer. Is there something you could divide by to achieve this result? Hint, look at the bottom left term!