simplify and evaluate $\frac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$ How do you simplify and evaluate $\dfrac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$?
What is the problem asking?

Hint:
Try to reduce this problem to one that makes use of the more familiar $\sin$ and $\cos$ sum of angle identities.
$$\begin{array}{lll}
\displaystyle\frac{\tan 80^\circ - \tan 20^\circ}{1+\tan80^\circ\tan20^\circ} &=&\displaystyle\frac{\tan 80^\circ + \tan (-20^\circ)}{1-\tan80^\circ\tan(-20^\circ)}\\
&=&\displaystyle\frac{\tan 80^\circ + \tan (-20^\circ)}{1-\tan80^\circ\tan(-20^\circ)}\cdot\frac{\cos 80^\circ\cos(-20^\circ)}{\cos 80^\circ\cos(-20^\circ)}\\
&=&\dots
\end{array}$$
Hint 2: $\tan 80^\circ\cos 80^\circ = ?$ 
You can take it from here I think.
