Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$ Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$?
The correct answer should be 4. 

You'd better to know
$$\sin 2\theta = 2\sin \theta \cos \theta$$
$$\sin \alpha-\sin \beta=
2\cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}$$
Then we have
\begin{align*}
  \tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ} &=
  \frac{\sin 9^{\circ}}{\cos 9^{\circ}}-
  \frac{\sin 27^{\circ}}{\cos 27^{\circ}}-
  \frac{\cos 27^{\circ}}{\sin 27^{\circ}}+
  \frac{\cos 9^{\circ}}{\sin 9^{\circ}} \\
  &= \frac{1}{\sin 9^{\circ} \cos 9^{\circ}}-
     \frac{1}{\sin 27^{\circ} \cos 27^{\circ}} \\
  &= \frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}} \\
  &= \frac{2(\sin 54^{\circ}-\sin 18^{\circ})}{\sin 18^{\circ} \sin 54^{\circ}} \\
  &= 2\frac{2\cos 36^{\circ} \sin 18^{\circ}}{\sin 18^{\circ} \cos 36^{\circ}} \\ 
  &= 4
\end{align*}
